HALL'S  MATHEMATICAL  SERIES 


COMPLETE 
/KRITAMETIC 

BY 


WERNER  SCMOOLBOOKCOMPAMY 

SiDUCATiOM.AU  PUBUSHESS 

NEW  YQRK;CMiCASO - BOSTO?!  _ 


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No.     3 


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AMERICAN  ^    ,.  ..u, 

A.  F.  GUNN,  L^i^nx  A^'t, 

204    PINE   S^-^-^TT, 
SAN   FRAN.._...^.. 


Digitized  by  tine  Internet  Archive 

in  2008  witii  funding  from 

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http://www.arcliive.org/details/completearitlimetOOIiallricli 


HALL'S  MATHEMATICAL  SERIES 


COMPLETE  ARITHMETIC 

ORAL  AND  WRITTEN 


BY 

FRANK  H.  HALL 

AUTHOR  OF   "THE  WERNER  ARITHMETICS,"  "THE  ARITHMETIC  READERS,'     ETC 


WERNER  SCHOOL  BOOK  COMPANY 

NEW  YORK        CHICAGO        BOSTON 


HALL'S  MATHEMATICAL  SERIES 


THE  WERNER  ARITHMETICS 

A  Three-Book  Course  for  Graded  Schools 

Book  I.  For  third  and  fourth  grades,  cloth,  256  pages,  400. 
Book  II.  For  fifth  and  sixth  grades,  cloth,  288  pages,  40c. 
Book  III.      For  seventh  and  eighth  grades,  cloth,  288  pages,  50c. 


TEACHERS'   HAND  BOOK 

giving  oral  work  preparatory  for  Book  I,  suggestions  to  teach- 
ers who  are  using  the  Werner  Arithmetics,  answers  to  proDiems 
in  Books  II  and  III,  and  a  large  amount  of  supplementary  seat- 
work.     Cloth,  131  pages,  25c. 


THE  HALL  ARITHMETICS 

A  Two- Book  Course  for  Graded  or  Ungraded  Schools 
Hall's  Elementary  Arithmetic,  cloth,  248  pages,  -         35c. 


Hall's  Complete  Arithmetic,  cloth,  448  pages, 


-     60c. 


IW 


■~'», 


Copyright,  1899 
By  WERNER  SCHOOL  BOOK  COMPANY 


Typography  by  R.  R.  Donnelly  Sons  Co..  Chicago 


^ 


PREFATORY  NOTE. 

In  Part  I.  of  this  book  (pp.  11-149),  classification  is  made 
subordinate  to  gradation.  Every  problem  is  selected,  not 
with  reference  to  the  place  it  occupies  in  a  scientific  classifica- 
tion of  mathematical  topics,  but  rather  with  reference  to  the 
supposed  thought-power  of  the  pupil.  But  systematic  arrange- 
ment of  the  matter  presented  is  not  ignored.  Seven  topics 
are  treated.  These  appear  on  the  first  ten  pages  of  the  book, 
and  each  topic  is  re-presented  in  each  ten-page  group.  Com- 
pare pages  15,  25,  35,  45,  etc.;  pages  17,  27,  37,  47,  etc. 

In  Part  II.  (pp.  151-369),  as  in  Part  I.,  each  page  is  a 
unit  of  the  greater  ten-page  unit.  The  first  six  pages  of  every 
ten-page  group  are  devoted  to  some  general  topic.  Upon  the 
seventh  and  eighth  pages  the  algebraic  phase  of  this  topic 
appears;  upon  the  ninth,  elementary  work  in  geometry,  and 
upon  the  tenth,  miscellaneous  problems.  This  arrangement 
makes  the  book  convenient  for  reference  and  review,  and,  it 
is  believed,  will  greatly  aid  the  pupil  in  properly  correlating 
his  own  mathematical  knowledge. 

In  Part  III.  (pp.  371-442),  the  matter  is  arranged  under 
four  general  heads,  viz.:  Denominate  Numbers,  Short  Meth- 
ods in  Multiplication  and  Division,  Practical  Approximations, 
and  Miscellaneous  Problems.  With  the  denominate  number 
tables,  many  practical  problems  in  measurements  are  pre- 
sented. The  "short  methods"  are,  for  pedagogical  reasons, 
placed  near  the  close  of  the  book.  The  miscellaneous  prob- 
lems include  many  typical  sets  of  "examination  questions" 
supplied  to  the  author  for  this  use.  Altogether,  Part  III. 
provides  for  a  complete  and  thorough  application  of  the  prin- 
ciples presented  in  Parts  I.  and  II.  F.  H.  H. 

Jacksonville,  IllixNois,  Marc  h,  1899. 


CONTENTS  — PART  I. 


"The  Foundation," 

Suggestions  to  Teachers, 

Simple  Numbers,    - 

Common  Fractions, 

Decimals, 

Denominate  Numbers, 

Measurements, 

Ratio  and  Proportion, 

Percentage, 

Percentage, 

Review  Problems, 

Miscellaneous  Problems, 


-      5-9 
10 

11,  21,  31,  41,  51,  61,  etc. 

12,  22,  32,  42,  52,  62,  etc. 

13,  2.^,  33,  43,  53,  63,  etc. 

14,  24,  34,  44,  54,  64,  etc. 

15,  25,  35,  45,  55,  65,  etc. 

16,  26,  36,  46,  56,  66,  etc. 

17,  27,  37,  47,  57,  67,  etc. 

18,  28,  38,  48,  58,  68,  etc. 

19,  29,  39,  49,  59,  69,  etc. 

20,  30,  40,  50,  60,  70,  etc 


THE  FOUNDATION. 

To  THE  Pupil. — Read  each  problem  and  (a)  tell  its  meaning,  (b; 
solve  it,  and  (c)  tell  the  suggested  number  story.  Do  this  until  you 
can  easily  give  the  meaning  of  all  problems  similar  to  these,  solve 
them,  and  tell  the  suggested  number  stories  without  reference  to  the 
notes  that  follow. 


8^  +  2^  =  * 

^- 

.^ 

= 

# 

6^X2  =  ^ 

f  +  }=(t^i. 

f-i 

\  = 

« 

) 

f  bu.  X  7  =  (3) 

6f  +  5J=V) 

6i- 

5i 

= 

(0 

12^X|=(8) 

2  bu.  3  pk. 

5bu. 

1 

pk. 

$|XJ=(13) 

2  bu.  2  pk. 

Ibu. 

3 

pk. 

12^X2J=06) 

(") 

(.2) 

12jx3  =  (i») 

12^X21.  =  (22) 

6^  --  2<Z!  =  * 

6^  -^  2  =  * 

4-J  =  W 

\ 

bu.  ^  4  =  (5) 

l-i  =  (») 

6| 

«yd.-2  =  (,o) 

7^^2i=(.4 

) 

5J  tons -5- 2  =  (15) 

2  ft.)46  ft.  (,  1 

0 

2)46  ft. 

2jft.)15ft.  (20) 

(18) 

2}  bu.)13i  bu. 

3)27 1  ft. 

(23)  (21) 

1.  I  plus  J,  means,  \  and  \,    I  can  change  thirds  and  fourths  to 

twelfths;  |  —  twelfths;  \  =  twelfths; twelfths  and 

twelfths  =  twelfths.    Story— Harry  played  ball  |  of  an 

hour,  and  ^^hide  and  seek''  J  of  an  hour;  in  all  he  played  

twelfths  of  an  hour.    (See  First  Book,  pages  125, 135, 145,  147, 149.) 

2.  I  minus  ^,  means,  f  less  J.     I  can  change  fourths  and  thirds  to 
twelfths.     I  =  twelfths;  y=  twelfths;  twelfths  less 

*  See  Elementary  Arithmetic,  pp.  89  and  153. 

5 


g  THE    FOUNDATION. 

■-. —  twelfths  = twelfths.    Story —William  had  a  piece  of  wire 

I  of  a  fcor.  long;  he  cut  from  it  apiece  \  of  a  foot  long;  what  re- 
mained was  twelfths  of  a  foot  long. 

3.  I  of  a  bushel  multiplied  by  7,  means,  take  7  times  three  fourth- 
bushels;   7  times  3  fourth-bushels  =  fourth-bushels,  or  

and bushels.    Story— A  farmer  fed  to  his  horses  |  hu.  of 

oats  each  day  for  a  week ;  in  all  he  fed and bushels. 

4.  4  divided  by  J,  means,  find  how  many  times  1  third  is  contained 
in  4.     I  can  change  4  to  thirds.    4  is  12  thirds.    1  third  is  contained 

in  12  thirds times.   Story — The  teacher  divided  4  oranges  among 

some  boys,  giving  to  each  ^  of  an  orange ;  there  were  boys. 

5.  i  of  a  bushel  divided  by  4,  means,  find  1  fourth  of  J  of  a  bushel. 

1  fourth  of  ^  of  a  bushel  = of  a  bushel.    Story — A  boy 

divided  \  of  a  bushel  of  oats  equally  among  4  ponies ;  each  pony 
received of  a  bushel. 

6.  6|  plus  5|,  means,  6|  and  5|.    6|  and  5|  = and . 

Story — In  one  jar  there  ivere  6^  lb-  of  butter ;  in  another  there  were 
5^  lb. ;  in  both  there  were  and  •  pounds. 

7.  6|  minus  5^,  means,  6|  less  5J.    6|  less  b^  -  - —  and -. 

Story — A  grocer  had  6^  lb.  cheese,  from  ivhich  he  sold  5^  lb. ;  there 
remained and pounds. 

8.  12^  multiplied  by  |,  means,  take  |  of  12^.    3  fourths  of  12^  = 

cents.    Story — At  12f  a  pound,  I  of  a  pound  of  meat  costs 

cents. 

9.  I  divided  by  ^,  means,  find  how  many  times  1  sixth  is  contained 
in  2  thirds.     I  can  change  thirds  to  sixths;  2  thirds  =  sixths; 

1  sixth  is  contained  in  4  sixths  times.     Story  —  Mrs.  Smith 

divided  ^  of  a  pie  am,ong  some  boys,  giving  to  each  J  of  a  pie ;  there 
were  boys. 

10.  6|  yd.  divided  by  2,  means,  find  1  half  of  6|  yards.    1  half  of 

6|  yards  =  and yards.     Story— A  salesman  cut  6'| 

yd.  of  ribbon  into  2  equal  pieces ;   there  were  and 

yards  in  each  piece. 

11.  2  bu,  3  pk.  plus  2  bu.  2  pk.,  means,  2  bu.  3  pk.  and  2  bu.  2  pk. 

2  bu.  3  pk.  and  2  bu.  2  pk.  =  bu. pk.    Story  —A  dealer  put 

2  bu.  3 pk.  of  oats  into  one  bag  and  2  bu.  2 pk.  into  another  bag;  in 
both  there  were  bushels  peck. 

12.  5  bu.  1  pk.  minus  1  bu.  3  pk.,  means,  5  bu.  1  pk.  less  1  bu.  3  pk. 

5  bu.  1  pk.  less  1  bu.  3  pk.  =  bu.  pk.     Story — Mr.  Bean 

had  5  bu.  1  pk.  of  oats,  from  which  he  fed  1  bu.  3  pk. ;  there  remained 
bushels  pecks. 

13.  $1  multiplied  by  J,  means,  take  1  half  of  3  fifths  of  a  dollar. 

1  half  of  3  fifths  of  a  dollar  = of  a  dollar.    Story— At  |  of 

a  dollar  a  yard,  1  half  of  a  yard  of  ribbon  is  worth of  a 

dollar. 


THE    FOUNDATION.  7 

14.  7^  divided  by  2|,  means,  find  how  many  times  2 J  is  contained 
in  7|.     I  can  change  2|  and  7|  to  halves;  2|  =  halves;  7|  = 

halves.     halves  are  contained  in  halves  times. 

Story — A  farmer  put  7J  bushels  of  oats  into  hags,  putting  2^  bushels 
in  each  bag ;  there  were  bags. 

15.  5f  tons  divided  by  2,  means,  find  1  half  of  5|  tons.    1  half  of 

5f  tons  =  and tons.      Story  —  5|  tons  of  coal  ivere 

divided  equally  between  two  families ;  each  family  received  

and tons. 

16.  12"?  multiplied  by  2|,  means,  take  2  times  12<?  plus  3  fourths 

of  12^  (2|  times  12?).    2  times  12«*  plus  3  fourths  of  12?  =  cents. 

Story — At  12^  a  pound  2%  lb.  of  cheese  cost  cents. 

17.  46  ft.  divided  by  2  ft.,  means,  find  how  many  times  2  ft.  are 

contained  in  46  ft.    2  ft.  are  contained  in  46  ft.  times.    Story — 

A  mechanic  cut  46  ft.  of  moulding  into  pieces,  each  piece  being  2 
ft.  long ;  there  were  pieces. 

18.  46  ft.  divided  by  2,  means,  find  1  half  of  46  ft.     1  half  of  46  ft. 

= ft.    Story— Henry  divided  46  ft.  of  wire  into  2  equal  parts ; 

each  part  was  feet  long. 

19.  12^  multiplied  by  3,  nieans,  take  3  times  12\  (3  times  \,  plus  3 

times  12).    3  times  12 ^  -  .    Story — One  side  of  a  triangle  having 

equal  sides  is  12\  ft. ;  the  perimeter  of  the  triangle  is  and  

feet. 

20.  15  feet  divided  by  2|  feet,  means,  find  how  many  times  2|  feet 
are  contained  in  15  feet.     I  can  change  15  ft.  and  1\  ft.  to  half-feet; 

15  ft.  =  half- feet;  2|  feet  =  half- feet;   half- feet  are 

contained  in  half-feet  times.    Story — A  mechanic  had  15 

feet  of  moulding  which  he  cut  into  pieces,  each  piece  being  2^  ft. 
long ;  there  were  pieces. 

21.  27 1  feet  divided  by  3,  means,  find  1  third  of  27^  feet.    1  third 

of  27^  feet  =  and feet.     Story  —The  perimeter  of  a 

triangle  having  equal  sides  is  27^  feet;  each  side  is  and  

feet. 

22.  12|  multiplied  by  2|,  means,  take  2  times  12^,  plus  J  of  12^. 

2  times  12^,  plus  i  of  12 J  =  and .    Story— At  $12\  per 

ton,  2^  tons  of  hay  cost  and dollars. 

.  23.  13|  bu.  divided  by  2^  bu.,  means,  find  how  many  times  2^  bu. 
are  contained  in  13|  bu.  I  can  change  2^  and  13^  to  fourths.  2^  = 
fourths.      13^  =  fourths.      fourths  are  contained  in 

fourths times.    Story — A  farmer  had  131  bu.  of  oats  which 

he  put  into  bags,  putting  2\  bu.  in  each  bag.     There  ivere bags. 


THE    FOUNDATION. 

.6  +  .2  =  *  .6  -  .2  =  *  .6X2=:* 

.24  +  .05  =  (i) .     .64  -  .05  =:  (2)      .12  x  9  ==  (3) 

.24  +  .5  =  (e)         .64  -  .5  =  (7)         20  x  -2  ==  (s) 

176.4  276.4  20x3.2  =  (13) 

148.75  148.23 


(11)  (12) 


.1  X.l  =  (i6) 
.3x.2  =  (i9) 


.6^.2  =  *  .6^2  =  * 

.08  ^  .02  =  (4)  $.08  ^  2  =  (5) 

8  ^  .4  =  (9)   ,  $6.36  -^  3  =  (1 0) 

4  -^  .05  =  (1 4)  5.6  tons  ^  4  =  (1 5) 
2.5  -^  .05  =  (1 1)  5)$24.5  (1  s) 

$.5)$24.5(2o)  5)$8.2    (21) 


1.  .24  plus  .05,  means,  24  hundredths  and  5  hundredths.    .24  and 

.05  = hundredths.    Story— William  had  $.24;  he  earned  4-05; 

he  then  had  . 

2.  .64  minus  .05,  means,  64  hundredths  less  5  hundredths.    .64  less 

.05  = hundredths.    Story—Martha  had  $.64;   she  spent  $.05; 

she  then  had . 

3.  .12  multiplied  by  9,  means,  take  9  times  12  hundredths.  9  times 
.12  =  hundredths,  or  and  hundredths. 

4.  .08  divided  by  .02,  means,  find  how  many  times  2  hundredths 

are  contained  in  8  hundredths.     .02  are  contained  in  .08  times. 

Story — I  paid  $.08  for  oranges  at  $.02  each ;  I  bought oranges. 

5.  $.08  divided  by  2,  means,  find  1  half  of  8  hundredths  of  a  dollar. 

One  half  of  $.08  =  ,    Story — I  paid  $.08  for  2  lemons ;  one  lemon 

cost . 

6.  .24  plus  .5,  means,  24  hundredths  and  5  tenths.    5  tenths  =:  50 

hundredths.      .24  and   .50  =  hundredths.      Story — John  had 

$.24;  Alfred  had  $.5;  together  they  had . 

7.  .64  minus  .5,  means,  64  hundredths  less  5  tenths.    5  tenths  =; 

50  hundredths.    .64  less  .50  =  hundredths.    Story — Sarah  had 

$.64 ;  Mary  had  $.5 ;  Sarah  had  $ more  than  Mary. 

*  See  Elementary  Arithmetic,  pp.  153  and  155. 


THE    FOUNDATION.  9 

8.  20  multiplied  by  .2,  means,  find  2  tenths  of  20.    One  tenth  of 

20  = ;  2  tenths  of  20  = .    Story— At  $20  an  acre,  .2  of  an 

acre  of  land  would  cost dollars. 

9.  8  divided  by  .4,  means,  find  how  many  times  4  tenths  are  con- 
tained in  8.    8  =  80  tenths.    4  tenths  are  contained  in  80  tenths 

times.    Story — I  paid  $8  for  potatoes  at  $A  {4  dimes)  a  bushel; 

I  bought  bushels. 

10.  $6.36  divided  by  3,  means,  find  1  third  of  $6.36.    One  third  of 

$6.36  =  .    Story— I  paid  $6.36  for  3  barrels  of  apples;  1  barrel 

cost . 

11.  The  sum  of  176.4  and  148.75  is  . 

12.  The  difference  of  276.4  and  148.23  is  . 

13.  20  multiplied  3.2,  means,  take  3  times  20,  plus  2  tenths  of  20. 

3  times  20  =  .    2  tenths  of  20  =  .    3  times  20,  plus  2  tenths 

of  20  =  .    Story — At  $20  an  acre,  3.2  acres  of  land  are  worth 

dollars. 

14.  4  divided  by  .05,  means,  find  how  many  times  5  hundredths 
are  contained  in  4.  4  =  400  hundredths.  5  hundredths  are  con- 
tained in  400  hundredths  times.    Story — I  paid  $4  for  tablets 

at  $.05  {5f)  each;  I  bought tablets. 

15.  5.6  tons  divided  by  4,  means,  find  1  fourth  of  5.6  tons.    One 

fourth  of  5.6  =  and tons.     Story — A  farmer  sold  4 

loads  of  hay,  the  entire  weight  of  the  hay  was  5.6  tons ;  the  loads 
averaged  and tons. 

16.  .1  multiplied  by  .1,  means,  find  1  tenth  of  1  tenth.  One  tenth 
of  1  tenth  = . 

17.  2.5  divided  by  .05,  means,  find  how  many  times  5  hundredths 
are  contained  in  2.5.  2.5  =  250  hundredths.  5  hundredths  are  con- 
tained in  2.5  (250  hundredths)  times.    Story — I  paid  $2.5  for 

pencils  at  $.05  (5^)  each;  I  bought  pencils. 

18.  $24.5  divided  by  5,  means,  find  1  fifth  of  $24.5.  One  fifth  of 
$24.5  =  .    Story — I  paid  $24.5  for  5  tons  of  coal;   1  ton  cost 

19.  .3  multiplied  by  .2,  means,  find  2  tenths  of  3  tenths.    One 

tenth  of  1  tenth  =  .    One  tenth  of  3  tenths  =  -.    Two  tenths 

of  3  tenths  =  . 

20.  $24.5  divided  by  $.5,  means,  find  how  many  times  5  tenths  of 
a  dollar  are  contained  in  $24.5  (245  tenths  dollars).  5  tenths  are  con- 
tained in  24.5  (245  tenths) times.    Stoi^—I paid  $24.5  for  apples 

at  $.5  (5  dimes)  a  bushel;  I  bought  bushels. 

21.  $8.2  divided  by  5,  means,  find  1  fifth  of  $8.2.    One  fifth  of 

$8.2  ($8.20)  =  .    Story— I  paid  $8.2  for  5  yards  of  cloth;  1  yard 

cost  . 


SUGGESTIONS  TO  TEACHERS. 


ORDER   OF    PROCEDURE   IN   PART   I. 

Step  1. — Prepare  the  pupil  by  means  of  oral  instruction  for  the 
work  of  a  given  page.  This  preparation  may  be,  in  part,  the  slow 
reading  to  the  pupil  of  the  figure  problems'^  upon  the  page,  the 
teacher  hesitating  at  each  blank  for  the  pupil  to  supply  the  word. 
In  this  preparatory  work,  no  book  should  be  in  the  hands  of  the 
pupil.  New  words  should  be  first  presented  through  the  voice  of  the 
teacher  in  the  expression  of  thought.  They  may  then  be  written 
upon  the  blackboard  by  the  teacher,  erased,  and  the  pupil  exercised 
in  both  oral  and  written  reproduction. 

Step  2. — The  book  should  be  put  into  the  hands  of  the  pupil,  and 
he  should  read,  first  silently,  then  orally,  the  figure  problems  upon 
any  page  for  which  proper  preparation  has  been  made. 

Step  3. — The  pupil  may  attempt  the  letter  problems*  at  his  desk 
without  further  assistance.  If  he  is  unable  to  solve  these,  review 
the  figure  problems  and  give  others  similar  to  them.  In  some  in- 
stances it  may  be  well  to  have  the  figure  problems  solved  upon  the 
blackboard  as  preparation  for  the  letter  problems. 

2. 

Wherever  the  word  ''story"  follows  a  problem,  as  on  pp.  12  and 
13,  require  the  pupil  to  make  a  statement  showing  how  the  problem 
might  originate  in  business  or  other  experience;  thus,  the  story  for 
problem  (f),  page  12,  might  be,  A  man  divided  4|  acres  of  land  into 
lots  each  containing  ^  of  an  acre.     There  were  95  lots. 

3. 

The  separatrix,  as  an  aid  in  "  pointing  off  "  in  multiplication  and 
division  of  decimals,  is  mentioned  in  a  note  on  page  133.  Its  use  is 
illustrated  on  page  143.  Its  occasional  use  in  blackboard  work 
earlier  in  the  course  may  be  helpful;  but  a  too  early  reliance  upon 
formal  rules  is  to  be  avoided  if  the  purpose  of  the  work  is  the 
development  of  the  thought-power  of  the  pupil. 

*  The  figure  problems  are  those  designated  by  figures ;  the  letter  problans,  those 
designated  by  letters. 

10 


ARITHMETIC, 


PART  I. 

SIMPLE    NUMBERS. 

1.  Three  is  an  exact  divisor  of  6,  of  9,  of  ,  of  . 

of  ,  and  of  .     Four  is  an  exact  divisor  of  . 

2.  Eighteen  is  exactly  divisible  by  2,  by  ,  by  , 

and  by  .     28  is  exactly  divisible  by  . 

3.  Two  is  an  exact  divisor  of  4,  6,  ,  ,  ,  etc. 

4.  A  number  that  is  exactly  divisible  by  2  is  called  an 
even  number;  8,  10,  ,  ,  ,  are  even  numbers. 

5.  Two  is  not  an  exact  divisor  of  1,  3,  5,  ,  ,  etc. 

6.  A  number  that  is  not  exactly  divisible  by  2  is  called 
an  odd  number;  9,  11,  ,  ,  ,  ,  are  odd  num- 
bers. 

7.  37  is  an  number.         24  is  an  number. 

8.  Tell  which  of  the  following  are  odd  numbers  and 
which  are  even  numbers:  375,  256,  320,  197,  281,  378, 
584,  252,  323,  569. 

(a)  Copy  the  numbers  given  in  problem  8  and  find  their 
sum.* 

(b)  Find  the  sum  of  all  the  odd  numbers  from  1  to  15 
inclusive. 

(c)  Find  the  sum  of  all  the  even  numbers  from  2  to  16 
inclusive. 

*  Problems  designated  by  letters  are  for  the  slate. 

11 


12  COMPLETE    ARITHMETIC. 

FRACTIONS. 

1.  When  two  or  more  fractions  have  denominators  that 
are  alike,  they  are  said  to  have  a  common  denominator. 
^1^,  ^,  ^5^,  and  y^g-  have  a .     f  and  f  do  not  have 


Reduce  to  equivalent  fractions  having  a  common  denominator. 

2.  h  h  i'  (Change  to  12ths.)  i=       i=       ^  = 

3.  h  h  h  (Change  to )  ^  =       i=       ^  = 

4.  },  |.  (Change  to )  |  =        \  = 

5.  ^,  4-  (Change  to )  |  =       |  = 

(a)  f  and  f              (b)  f  and  f  (c)  |  and  |. 


I  can  change  eighths  and  thirds  to 

ths.      f  =  ^T-      i  =  YT'     twenty-fourths  and  

twenty-fourths  = twenty-fourths. 

(d)  Find  the  sum  of  576|-  and  239i. 

7.  From  4  subtract  1     I  can  change  sevenths  and  thirds 

to  sts.     i  =YT'     i  =  YT-      twenty-firsts  less  

twenty-firsts  =  twenty-firsts. 

(e)  Find  the  difference  of  572f  and  368i 

8.  Divide  |-  by  -^^.     This  means,  find  how  many  times 
■gig-   is   contained   in   |-.      I    can    change    |-    to    twentietlis. 

3.  =  .^^,     1  twentieth  is  contained  in  twentieths   

times.     Story* 

(f)  Find  the  quotient  of  4|-  divided  by  ^q.     (Change  4| 
to  twentieths.)     Story. ^ 

(g)147f  +  38|.    (h)374f-18H.    (i)  27ibu. -^  2^bu4 

*  See  Note  (9)  page  6.  t  See  paragraph  2,  page  10. 

I  Tell  the  meaning.     Change  27J  and  2\  to  lialves.     Story.     See  note  (14)  page  7. 


PART    I.  13 


DECIMALS — THOUSANDTHS. 


1.  .125  and  .006  are  thousandths. 

2.  .125  and  .06  are  thousandths. 

3.  .125  and  .6  are  thousandths. 

4.  .275  less  .006  are  thousandths. 

5.  .275  less  .06  are  thousandths. 

6.  2.4  and  3.104  are  and  thousandths. 

(a)  Add  27.006,  14.8,  and  25.06. 

7.  36.025  less  1.008  equals  and  thousandths. 

(b)  From  8.175  subtract  3.023. 

8.  3  times  2.005  is  and  thousandths. 

(c)  Multiply  26.008  by  4. 

9.  Divide  .035  by  .007.     This  means  4*  . 

(d)  Find  the  quotient  of  .375  divided  by  .005.    Story.^ 

10.  Divide  .035  by  7.     This  means  5*  — -. 

(e)  Find  the  quotient  of  35.049  acres  divided  by  7.    Story. 

11.  At  $.05  each,  for  $1.25  I  can  buy  pencils.    1.25 

-^.05  = 

12.  At  $.05  each,  for  $2  I  can  buy  pencils.    2  -  .05  = 

(f)  (g)  (h)  (i)  (J) 

Add.         Subtract.     Multiply.  Divide.  Divide. 

46.27  54.75  7.054:i:        $.005)$.385§        5)$.385§ 

25.308  8.326        5 

♦These  figures  refer  to  notes  on  page  8. 

t  See  page  10,  paragraph  2. 

X  Write  the  decimal  point  in  the  product  immediately  after  writing  the  tenths'  figure 
of  the  product. 

§"  Point  off"  in  division  of  decimals  by  thinking  what  the  problems  mean. 
Problem  (i)  means,  find  how  many  times  5  thousandths  arc  contained  in  385  thousandths. 
5  thousandths  are  contained  in  383  thousandths  77  times.  Problem  (j)  means, ^wd  1 
fifth  of  385  thousandihs.    J  of  385  thousandths  is  77  thousandths,  or  .077. 


14  COMPLETE    ARITHMETIC. 

DENOMINATE    NUMBERS. 
2000  lb.  =  1  ton. 

1.  500  lb.  are of  a  ton. 

2.  1600  lb.  are of  a  ton. 

3.  1000  lb.  are of  a  ton. 

4.  200  lb.  are of  a  ton. 

6.  I  weigh  pounds. 

6.  My  father's  horse  weighs  pounds. 

7.  A  keg  of  nails  weighs  100  pounds. 

8.  One  ton  of  nails  is  kegs. 

9.  A  load  of  hay  weighs  from  to  pounds.* 

10.  A  load  of  coal  weighs  from  to  pounds.* 

11.  When  hay  is  $12  a  ton,  1000  lb.  cost  . 

12.  When  coal  is  $8  a  ton,  1500  lb.  cost  ■ . 

13.  When  meal  is  $16  a  ton,  500  lb.  cost . 

14.  When  hay  is  $6  a  ton,  200  lb.  cost  — . 

15.  When  hay  is  $7  a  ton,  200  lb.  cost . 

16.  When  hay  is  $5  a  ton,  200  lb.  cost . 

17.  When  hay  is  $8  a  ton,  200  lb.  cost . 

18.  When  hay  is  $6  a  ton,  2200  lb.  cost  . 

19.  Wlien  hay  is  $8  a  ton,  2200  lb.  cost  . 

20.  When  hay  is  $5  a  ton,  2200  lb.  cost . 

21.  2500  lb.  are  and tons. 

22.  3000  lb.  are  and tons. 

23.  4500  lb.  are  and tons. 

24.  5000  lb.  are  and tons. 

25.  When  hay  is  $12  a  ton,  2500  lb.  cost . 

26.  When  hay  is  $10  a  ton,  3000  lb.  cost . 

27.  When  hay  is  $8  a  ton,  4500  lb.  cost  . 

♦When  you  see  a  load  of  hay  or  a  load  of  coal,  courteously  ask  the  man  in 
charge  of  it  to  tell  you  how  much  it  weighs. 


PART    1. 

MEASUREMENTS. 


15 


1-inch  cube. 

1.  A  2-inch  square  contains  square  inches. 

2.  A  2 -inch  cube  contains  cubic  inches. 

3.  A  3-inch  square  contains  square  inches. 

4.  A  3 -inch  cube  contains  cubic  inches. 


5.  A  2 -foot  square  contains 

6.  A  2 -foot  cube  contains  - 

7.  A  3-foot  square  contains 

8.  A  3-foot  cube  contains  - 


-  square  feet, 
cubic  feet. 

-  square  feet, 
cubic  feet. 


9.  The  area  of  a  2 -inch  square  is 

10.  The  soUd  content  of  a  2 -inch  cube  is 

11.  The  area  of  a  3-inch  square  is 

12.  The  solid  content  of  a  3-inch  cube  is 


inches. 


inches. 


13.  The  area  of  a  2-foot  square  is feet. 

14.  The  soHd  content  of  a  2-foot  cube  is  . 

15.  Find  the  area  of —  Find  the  sohd  content  of — * 

(1)  A  4-inch  square.         (a)  A  4-inch  cube. 


(2)  A  5 -inch  square. 

(3)  A  6 -inch  square. 

(4)  A  12-inch  square. 


(b)  A  5 -inch  cube. 

(c)  A  6 -inch  cube. 

(d)  A  1 2 -inch  cube. 


♦Problems  designated  by  letters  are  for  the  slate. 


16  COMPLETE    ARITHMETIC. 

RATIO   AND    PROPOKTION. 

1.  One  fourth  of  20  is  .  20  is  ^  of  — 

2.  One  half  of  11  is  .  11  is  i  of  — 

3.  One  third  of  10  is  .  10  is  i  of  — 

4.  One  fifth  of  45  is  .  45  is  i  of  — 

5.  Two  thirds  of  60  are  .  60  is  -|  of  — 

6.  Two  thirds  of  36  are  .  36  is  |  of  — 

7.  Two  thirds  of  15  are  .  15  is  |  of  — 

8.  Two  thirds  of  21  are  .  21  is  |  of  — 

9.  10  is of  15.     15  is 1 of  10,  or— times  10. 

10.  15  is of  20.     20  is of  15,  or  —  times  15. 

11.  20  is of  25.     25  is of  20,  or  —  times  20. 

12.  25  is of  30.     30  is of  25,  or  —  times  25. 

13.  Three  fourths  of  16  are  2  thirds  of  — . 

14.  Two  thirds  of  15  are  1  half  of  . 


15.  Three  fourths  of  24  are  2  thirds  of 

16.  Two  thirds  of  12  are  1  third  of  — 


17.  Eight  is of    12.      A  man   can   earn  

as  much  in  8  days  as  he  can  earn  in  12  days.     If  he 

can  earn  $21  in  12  days,  in  8  days  he  can  earn dollars. 

18.  Fifteen  is  and times  6.     A  man  can 

earn  times  as  much  in  15  days  as  he  can  earn  in  6 

days.     If  he  can  earn  $10  in  6  days,  in  15  days  he  can 
earn  dollars. 

(a)  If  a  man  can  earn  $104  in  6  weeks,  how  much  can 
he  earn  in  15  weeks. 

*It  may  be  well,  after  the  pupil  has  solved  "  mentally  "  such  problems  as  Nos.  5 
and  6,  to  require  him  to  solve  them  with  a  pencil— this,  as  a  preparation  for  similar 
problems  in  which  larger  numbers  are  used. 

t  Allow  the  pupil  to  say,  "  15  is  3  halves  of  10,"  but  remind  him  that  the  expres- 
sion "3  halves  of  10,"  means,  5  times  1  haJfoflO. 


PART    I.  17 

PERCENTAGE* 

50  per  cent  =  .50  =  ^. 
25  per  cent  =  .25  =  ^. 
33iper  cent  =  .33L  =  -J. 
20  per  cent  =  .20  =  f 

1.  50  per  cent  of  10  =  10  is  50%  of  • . 

2.  25  per  cent  of  12  =  12  is  25%  of  . 

3.  33i  per  cent  of  12  ==  12  is  33^%  of  . 

4.  20  per  cent  of  10  =  10  is  20%  of  . 

5.  50  per  cent  of  8  =  8  is  50%  of  . 

6.  25  per  cent  of  8  =  8  is  25%  of  . 

7.  20  per  cent  of  15  =  15  is  20%  of  . 

8.  33|  per  cent  of  15  =  15  is  33^%  of  . 

(5)t  (5)t 

9.  5  is  per  cent  of  10.         5  is  %  of  20. 

10.  5  is  per  cent  of  15.  5  is  %  of  25. 

11.  4  is  per  cent  of  8.  4  is  %  of  12. 

12.  4  is  per  cent  of  16.  4  is  %  of  20. 

13.  3  is  per  cent  of  6.  3  is  %  of  15. 

14.  Henry  had  25  cents;  he  spent  20  per  cent  of  his 
money;  he  spent  cents. 

15.  Peter  spent  10  cents;  this  was  25  per  cent  of  all  he 
earned;  he  earned  cents. 

16.  Eoscoe  earned  60  cents  and  spent  30  cents;  he  spent 
per  cent  of  what  he  earned. 

*To  THE  Teacher. — Frequently  review  this  page,  and  give  many  similar  oral 
problems  as  a  preparation  for  page  27. 

fThese  figures  designate  the  percentage  cases  to  which  the  problems  below  them 
belong. 


18  COMPLETE    ARITHMETIC. 

PERCENTAGE. 
(1) 

1.  50%  of  18  is  .  50%  of  19  is  - 

(a)  Find  50  per  cent  of  724 ;  (b)  of  725. 

2.  25%  of  16  is  .  25%  of  17  is  - 

(c)  Find  25  per  cent  of  896 ;  (d)  of  897. 

3.  33-1%  of  18  is  .  33i%  of  19  is 

(e)  Find  331-  per  cent  of  726  ;  (f)  of  727. 

4.  20%  of  25  is  .  20%  of  26  is  - 

(g)  Find  20  per  cent  of  875  ;  (h)  of  877. 

(2) 

5.  7  is  50%  of  .  7^  is  50%  of  . 


(i)  376  is  50%  of  what?       (j)  376 1-  is  50%  of  what? 

6.  9  is  25%  of  .  9^-  is  25%  of  . 

(k)  524  is  25%  of  what  ?       (1)  524i-  is  25%  of  what  ? 

7.  8  is  33i%  of  .  8^  is  33^%  of  . 

(m)  652  is  33a%  of  what  ?     (n)  652 1-  is  33^%  of  what  ? 

(S) 

11  is  %  of  33. 

11  is  %  of  55. 

30  is  %  of  60. 


8.  11  is  - 

%  of  22. 

9.  11  is  - 

— %  of  44. 

10.  15  is  - 

%  of  30. 

(i) 

11.  One  per  cent  (1%)  of  a  number  is  1  hundredth  of 
the  number ;  2  %  is  2  hundredths ;  3  %  is  3  hundredths. 

12.  One  per  cent  of  500  is  .        2%  of  500  is  . 

13.  One  per  cent  of  600  is  .        3%  of  600  is . 


PART    1.  19 


REVIEW. 


1.  Tell  which  of  the  following  are  odd  numbers  and 


which  are  even  numbers:  14,  17,  19,  20,  24,  27. 

2.  The  fractions  y^y,  -^-g,  and  \\  have  a  denomi- 
nator. 

3.  Change  |  and  |-  to  equivalent  fractions  having  a  com- 
mon denominator.     I  can  change  thirds  and  fifths  to  . 

2  _        4   _ 

3  -       y  - 

4.  Add  -|  and  i.     I  can  change  7ths  and  3ds  to  . 

I   =   -g-j-.     i  =  -g^.      twenty-firsts  and  twenty- 
firsts  =  twenty-firsts. 

5.  Divide  |-  by  ^-^.     This  means,  find  how  many  times 

^  is  contained  in  f .     I  can  change  5ths  to  .     |  =  ^-g-. 

One  twentieth  is  contained  in  twentieths  times. 

6.  Multiply  6.05  by  3.     Three  times  . 

(a)  Find  the  product  of  34.02  multipHed  by  4. 

7.  Divide  $6.28  by  2.     This  means,  10*  . 

(b)  Find  the  quotient  of  $475.65  divided  by  5. 

8.  When  hay  is  $12  a  ton,  1500  lb.  cost  dollars. 

9.  When  coal  is  $7  a  ton,  3000  lb.  cost  dollars. 

10.  A  2 -inch  square  is  how  many  times  as  large  as  a 
1-inch  square  ? 

11.  A  2  inch  cube  is  how  many  times  as  large  as  a  1-inch 
cube  ? 

(c)  How  many  square  inches  in  an  oblong  9  inches  by  23 
inches  ? 

12.  Three  fourths  of  20  are  1  half  of  . 

(d)  Three  fourths  of  84  are  1  half  of  what  number  ? 

♦See  note  10,  pag^ft 


20  COMPLETE    ARITHMETIC. 

MISCELLANEOUS. 

1.  Mr.  A  owned  15^  acres  of  land;  he  sold  6^  acres;  ha 
then  had  acres. 

(a)  Mr.  B  owned  546^  acres  of  land;  he  sold  228^  acres. 
How  many  acres  had  he  remaining  ? 

2.  A  bushel  of  oats  weighs  32  lb.;  2  bushels  of  oats 
weigh  pounds;  3  bushels  weigh  pounds. 

(b)  Sixty  bushels  of  oats  weigh  how  many  pounds  less 
than  one  ton  ? 

3.  Two  pounds  of  coffee  @  $.23  a  pound  cost  . 

(c)  Find  the  cost  of  84  pounds  of  coffee  @  $.23  a  pound.* 

4.  From  a  piece  of  cloth  containing  12  yards  there  were 
sold  5|-  yards  and  2^  yards;  there  were  left  yards. 

(d)  From  a  piece  of  cloth  containing  55  yards,  there 
were  sold  24|-  yd.  and  17|-  yd.    How  many  yards  were  left? 

5.  A  boarding-school  uses  one  gallon  of  milk  each  day ; 
at  5^  a  quart  the  milk  for  one  week  costs  . 

(e)  A  boarding-school  uses  4  gal.  of  milk  each  day;  at 
5^  a  quart  how  much  will  the  milk  for  the  month  of  October 
cost  ? 

6.  John  bought  20  oranges  at  3^  each  and  sold  them  at 
6f  each ;  he  gained  cents. 

(f)  John's  father  bought  25  bushels  of  potatoes  at  32^  a 
bushel  and  sold  them  at  45^  a  bushel.  How  much  did  he 
gain? 

*In  multiplication  of  decimal  fractions,  require  the  pupil  to  locate  the  decimal 
point  in  the  product  immediaMy   after  he  has  written  the  figure  to  the 

right  of  the  decimal  point  in  any  whole  or  pjirtial  product.    Thus,  in  the  ^-23 

multiplication  of  $.23  by  84  he  writes  (and  thus  locates)  the  decimal  point  of     

the  product  immediately  after  writing  the  figure  9  in  the  first  partial  pro-  ^ 

duct.    This  will  be  quite  clear  to  the  pupil  if  he  understands  that  he  first  — '- — 

finds  4  times  .23;  then  80  times  .23;  then  the  sum  of  these  two  partial  519'32 
products. 


PART    1.  21 

SIMPLE    NUMBERS. 

1.  Seventeen  is  an  integral  number. 

2.  Five  eighths  is  a  fractional  number. 

3.  Seventeen  and  five  eighths  is  a  mixed  number. 

4.  25  is  an  number.         5|-  is  a  number 

5.  |-J-  is  a  number.  3.5  is  a  ■  number. 

6.  .8  is  a  number.  18  is  an  number. 

7.  A  number  is  exactly  divisible  by  2  if  the  right-hand 
figure  is  0,  2,  4,  6,  or  8.  Tell  which  of  the  following  are 
exactly  divisible  by  2 :  241,  136,  274,  393,  247,  826. 

8.  A  number  is  exactly  divisible  by  5  if  its  right-hand 
figure  is  0  or  5.  Tell  which  of  the  following  are  exactly 
divisible  by  5  :  184,  275,  320,  145. 

9.  The  following  are  exactly  divisible  by  2  :  16,  38,  54, 
68,  ,  ,  ,  . 

10.  The  following  are  exactly  divisible  by  5 :  85,  140, 
175,  180,  ,  ,  ,  . 

11.  The  following  are  exactly  divisible  by  10:  30,  40, 
80,  120,  ,  ,  ,  . 

12.  There  are  twice  as  many  5's  as  there  are  lO's  in  a 
number.     90  is  tens.     90  is  fives. 

(a)  Divide  $2150  by  50.*  (b)  Divide  $215.00  by  50. 

(c)  Divide  $2295  by  51.  (d)  Divide  $290.70  by  61. 

(e)  Divide  $2058  by  49.  (f)   Divide  $156.80  by  49. 

(g)  Divide  $3276  by  52.  (h)  Divide  $322.40  by  52. 

(i)   Divide  $2496  by  48.  (j)    Divide  $254.40  by  48. 

*  This  means,  find  1  fiftieth  of  §2150.    1  fiftieth  of  $2150  is dollars.    Story- 

If  50  acres  of  land  cost  $2150, 1  acre  cost . 

t  In  solving  such  problems  as  (b),  (d),  etc.,  require  the  pupil  to  write  the  deci- 
mal point  in  the  quotient,  immediately  after  writing  the  units'  figure  of  the  quotient 
If  the  pupil  thinks  what  the  problem  means,  he  will  easily  "  point  off "  correctly. 


22  COMPLETE    ARITHxAIETIC. 

COMMON   FRACTIONS. 
Reduce  to  equivalent  fractions  having  a  common  denominator : 

1.  I  and  ^.      (Change  to  .)        |  =  -i-  = 

(a)  I  and  f .         (b)  f  and  |.         (c)  f  and  -J. 

2.  Add  I  and  f      (Change  to  ths.)* 

(d)  Find  the  sum  of  248 1  and  467f 

3.  From  |-  subtract  |.     (Change  to  ths.)-|- 

(e)  Find  the  difference  of  837}  and  284|. 

4.  Divide  6  by  |.  This  means,  find  how  many  times  ^ 
are  contained  in  6.  I  can  change  6  to  thirds.  6  =  -g .  2 
thirds  are  contained  in  thirds  times.     Story.-^ 

(f)  Find  the  quotient  of  48  divided  by  |.     Story.^ 

5.  Divide  8|-  feet  by  2.  This  means,  find  1  half  of  8|- 
feet     1  half  of  8^  feet  ^         Story. 

(g)  Find  the  quotient  of  86^  miles  divided  by  2.    Story. \^ 

(h)  3^  miles)154  miles.      This  means,  find  how  many  times 

3\  miles  are  contained  in  154  miles. 

(Change  3^  and  154  to  halves.)     Story — A  canal-boat  moved  at  the 

rate  of  3^  miles  an  hour;  to  move  154  miles  would  require  

hours. 

(i)  3)1654-  miles.       This  means,  find  1  third  of  1651  niiles. 
Story  —  A   train   moved   165^   miles   in   3 
hours ;  it  moved  at  the  rate  of miles  an  hour. 

*  See  page  12,  problem  6. 
t  See  page  12,  problem  7. 
t  See  note  (4),  page  6. 

g  48  feet  of  ribbon  was  cut  into  pieces  5  of  a  foot  long ;  there  were pieces. 

I  A  train  moved  86J  miles  in  2  hoiirs ;  this  was  at  the  rate  of miles  an  hour. 


PART    I.  23 


DECIMAL    FRACTIONS. 

1.  One  thousandth  of  a  dollar  is  mill. 

2.  Four  thousandths  of  a  dollar  are  4  . 

3.  Four  hundredths  of  a  dollar  are  4  . 

4.  Four  tenths  of  a  dollar  are  4  . 


5.  .1  of  $60  = 

.1  of  $4  = 

.1  of  $64  =* 

6.  .1  of  $.1  =. 

.1  of  $.5  = 

.1  of  $.7  = 

7.  .1  of  $6  = 

.1  of  $8.5  = 

.1  of  $5.3  = 

8.  .1  of  $.01  = 

.1  of  $.03  = 

.1  of  $.07  = 

9.  .1  of  $6.42  = 

.1  of  $7.56  = 

.1  of  $8.42  := 

10.  .1  of  $875  = 

.1  of  $74.2  = 

.1  of  $5.35  = 

(a)  Multiply  $374  by 

.3.   This  means. 

,  find  3  tenths  of  3 

Operation. 

$374t 
.3 

Explanation. 
One  tenth  of  $374  = 

Three  tenths  of  $374  = 

$112.2 

NUMBER   STORY. 

If  one  acre  of  land  is  worth  $374, 

1  tenth  of  an  acre  of  land  is  worth  . 

3  tenths  of  an  acre  of  land  are  worth  . 

(b)  (0)  (d)  (e)  (f) 

Multiply.  Multiply.  Multiply.  Divide.  Divide 

346  23.5  2.73  $.05)$.85$  5)$.85§ 

.3  .3  3  " 


*  By  means  of  problems  5  to  10  and  other  similar  exercises,  lead  the  pupil  to  see 
that  he  can  obtain  1  tenth  of  an  integral  number  by  "pointing  off "  one  figure  on 
the  right,  and  of  a  mixed  decimal  by  removing  the  point  one  place  to  the  left. 

tTh(j  pupil  should  understand  that  he  multiplies  S37.4  (not  $374)  by  3,  and 
should  be  taught  to  write  the  decimal  point  in  the  product  immediately  after  wnting 
the  tenths'  figure  of  the  product.    See  page  10,  paragraph  3. 

t  This  mesins,  find  how  many  times  5  hundredths  are  contained  in  85  hundrcd'hs. 

I  This  means,  ^nrf  t  fifth  of  85  hundredths. 


24  COMPLETE    ARITHMETIC. 


DENOMINATE    NUMBERS. 


A  bushel  of  oats  weighs  32  lb. 

A  bushel  of  wheat  weighs  60  lb. 

A  bu.  of  corn  (shelled)  weighs  56  lb. 

A  bu.  of  corn  (not  shelled)  weighs  70  lb.* 

1.  Two  bushels  of  oats  weigh lb.;  5  bu.  weigh 

(a)  How  much  more  than  1  ton  do  75  bu.  of  oats  weigh 

2.  Two  bu.  of  wheat  weigh  ;  3  bu.  weigh  . 


(b)  Seventy-five  bushels  of  wheat  weigh  how  much  more 
than  two  tons  ?     (c)  How  much  less  than  three  tons  ? 

3.  Two  bushels  of  shelled  corn  weigh  ;  3  bu.  weigh 


(d)  Forty-five  bushels  of  shelled  corn  weigh  how  much 
more  than  1^  tons  ?     (e)  How  much  less  than  1^  tons  ? 

4.  Two  bushels  of  corn   (not  shelled)  weigh  ;    3 

bushels  weigh  ;  4  bushels  weigh  . 

(f)  How  much  more  than  three  fourths  of  a  ton  do  25 
bushels  of  corn  (unshelled)  weigh  ?  (g)  How  much  less 
than  one  ton  ? 

5.  One  hundred  pounds  of  oats  are  bushels  and 

pounds,  or  and  32nds  bushels. 


(h)  Five  hundred  pounds  of  oats  are  how  many  bushels  ? 
(i)  How  many  bushels  in  one  and  one  half  tons  of  wheat  ? 
(j)  How  many  bushels  in  two  tons  of  unshelled  corn  ? 

(k)  (1)  (m)  (n) 

32  lb.)768  lb.     32  lb.)769  lb.     32  lb.)770  lb.     32  lb.)772  lb. 


*  Explain  to  the  pupil  that  the  expression  "  a  bushel  of  corn  (not  shelled)," 
meau3,  the  amount  0/ unshelled  com  required  to  maHe  I  Ifushel  0/  shelled  com. 


PART    I. 

MEASUREMENTS. 


25 


^ 


.^y    /    ^ 

i 

1 

1 

1; 

I 

1.  A  3-foot  square  (or  its  equivalent)  is  called  a  square 
yard.     It  contains  square  feet. 

2.  A  3-foot  cube  (or  its  equivalent)  is  called  a  cubic 
yard.     It  contains  cubic  feet. 

3.  The    area    of   a   4-foot   square  is    square   feet, 

or  and square  yards. 


or 


or 


4.  The  solid  content  of  a  4-foot  cube  is  cubic  feet, 

(a)  How  many  cubic  yards  in  a  4-foot  cube  ? 

5.  The    area    of  a  5 -foot   square  is   square    feet. 

and square  yards. 

6.  The  solid  content  of  a  5 -foot  cube  is  cubic  feet. 

(b)  How  many  cubic  yards  in  a  5 -foot  cube  ? 


The   area  of   a 
-  square  yards. 


6-foot   square  is 


square    feet. 


cubic  feet. 


8.  The  solid  content  of  a  6-foot  cube  is  - 
(c)  How  many  cubic  yards  in  a  6-foot  cube  ? 

9.  Find  the  area  of —  Find  the  solid  content  of — 

(1)  A  7-foot  square.  (d)  A  7-foot  cube. 

(2)  An  8-foot  square.  (e)  An  8-foot  cube. 

(3)  A  9-foot  square.  (f)  A  9-foot  cube, 


26  COMPLETE    ARITHMETIC. 

KATIO   AND   PROPORTION. 

1.  One  fifth  of  25  is  .  25  is  |  of  . 

2.  One  fourth  of  21  is  .  21  is  |  of  -. 

3.  One  third  of  22  is  .  22  is  |  of  . 

4.  One  half  of  45  is  .  45  is  i  of  . 

5.  Three  fourths  of  24  are  .  24  is  |  of  . 

6.  Three  fourths  of  36  are  .  36  is  |-  of  — — . 

(b)  96  is  f  of  what  number? 
(d)  96  is  f  of  what  number? 
(f)  96  is  1^  of  what  number? 

of  18.     18  is of  12. 

of  24.     24  is of  18. 

of  30.     30  is of  24. 

10.  Two  thirds  of  18  are  1  half  of  . 

11.  Three  fourths  of  24  are  2  thirds  of  . 

12.  Two  thirds  of  30  are  1  half  of  . 

(g)  One  third  of  132  is  1  half  of  what  number? 

(h)  Two  thirds  of  252  are  1  half  of  what  number? 

(i)  Three  fourths  of  96  are  2  thirds  of  what  number? 

(j)  Two  thirds  of  96  are  3  fourths  of  what  number? 

13.  Nine   is of    12.      A   man    can    earn  

—  as  much  in  9  days  as  he  can  earn  in  12  days.     If  he 


(a) 

Findf 

of  96. 

(c) 

Find  1 

of  96. 

(e) 

Find^ 

of  96. 

7. 

12  is  - 

8. 

18  is  - 

9. 

24  is  - 

can  earn  $40  in  12  days,  in  9  days  he  can  earn dollars, 

(k)  If  a  man  can  earn  $896  in  12  months,  how  many  dol- 
lars can  he  earn  in  9  months  ? 

(1)  If  Mr.  Conrad's  horses  consume  726  bushels  of  oats 
in  a  year,  how  many  bushels  will  be  required  to  feed  them 
8  months  ? 


PART    I.  27 


PERCENTAGE. 

16|  per  cent  =  .161  = 
14f  per  cent  =  .14|  = 
12^per  cent  =  .121-  = 


{1)  {2) 

1.  16f  per  cent  of  12  =  12  is  16 1%  of  — 

2.  14f  per  cent  of  21  =  21  is  14f  %  of  — 

3.  12|-  per  cent  of  16  =  16  is  12|-%  of  — 

4.  16f  per  cent  of  18  =  18  is  16|%  of  — 

5.  14|  per  cent  of  28  =  28  is  14f  %  of  — 

6.  12|-  per  cent  of  24  =  24  is  12|-%  of  — 

7.  50  per  cent  of  12  =  12  is  50%  of  . 

8.  25  per  cent  of  16  =  16  is  25%  of  . 

9.  20  per  cent  of  25  =  25  is  20%  of  . 

10.  331-  per  cent  of  18  =  18  is  33^%  of  - — . 

(5)  (5) 

11.  2  is  per  cent  of  6.  2  is  %  of  4. 

12.  2  is  per  cent  of  8.  2  is  %  of  14. 

13.  2  is  per  cent  of  12.  2  is  %  of  10. 

14.  2  is  per  cent  of  16.  1  is  %  of  2. 

15.  There  were  40  pears  on  a  tree;  25%  of  them  fell  off; 
pears  remained  on  the  tree. 

16.  William  sold  7  melons;  these  were  12|-%  of  all  the 
melons  h3  raised ;  he  raised  melons. 

17.  Mary  had  30  little  chickens;  a  hawk  killed  five  of 
them ;  the  hawk  killed  . per  cent  of  her  chickens. 

18.  Twelve  and  one  half  per  cent  of  $48  is  dollars 

(a)  Find  12i-  per  cent  of  $992. 

(b)  Find  16|  per  cent  of  $852. 


J8  COMPLETE    ARITHMETIC. 

PERCENTAGE. 
(1) 

1.  16|%  of  24=  16f%  of  25  = 
{Si)  Find  16|  per  cent  of  342  ;  (b)  of  343. 

2.  12|-%  of  24  =  121-%  of  25  = 
(c)  Find  12|-  per  cent  of  976  ;  (d)  of  978. 

3.  142%  of  28  =  14f  %  of  30  = 
(e)  Find  14|  per  cent  of  994;  (f)  of  996. 

(2) 

4.  5  is  121-%  of  .  5i  is  12J%  of  . 

(g)  246  is  12|-%  of  what  ?      (h)  246^  is  12^-%  of  what? 

5.  5  is  14f  %  of  .  5J  is  142%  of  . 

(i)  351  is  14f%  of  what?      (j)  356|  is  14f  %  of  what? 

6.  5  is  16f  %  of  .  5^  is  16f  %  of  . 

(k)  239  is  16f  %  of  what?      (1)  241^  is  16|%  of  what? 

(3) 

7.  12  is  %  of  24.  12  is  %  of  36. 

8.  12  is  %  of  48.  12  is  %  of  60. 

,  9.  12  is  %  of  72.  12  is  %  of  84. 

10.  12  is  %  of  96.  15  is  %  of  45. 

(1) 

11.  One  per  cent  of  $300  is  .  2%  of  $300  = 

12.  One  per  cent  of  $320  is  .  2%  of  $320  = 

13.  One  per  cent  of  $325  is  .  2%  of  $325  = 

14.  One  per  cent  of  $342  is  .  2%  of  $342  = 

(m)  Find  3%  of  $342.  (n)  Find  4%  of  $342. 

(o)  Find  5%  of  $342.  (p)  Find  6%  of  $342. 

(q)  Find  3%  of  $536.  (r)  Find  4%  of  $536. 


PART    I.  29 

REVIEW. 

1.  Two  is  an  exact  divisor  of  28,  274,  ,  . 

2.  Five  is  an  exact  divisor  of  75,  230,  ,  . 

3.  Ten  is  an  exact  divisor  of  80,  ,  ,  . 

4.  Twenty  is  an  integral  number.     |-  and  .5  are  

numbers.     5\  and  3.2  are  numbers. 

5.  The    fractions   -^^,  -^q,  ,  ,  have  a  common 

denominator.      |  and  do  not  have  a  common  denonii- 

liator. 

6.  The  sum  of  .135  and  .6  is  .         .135  -  .06  = 

7.  Divide  .045  by  .009.     (This  means  4*  .) 

(a)  Find  the  quotient  of  .875  divided  by  .005. 

8.  Divide  .045  by  9.     (This  means  5*  .) 

(b)  Find  the  quotient  of  54.063  divided  by  9. 

9.  When  hay  is  $8  a  ton,  2500  lb.  cost  . 

10.  Five  bushels  of  wheat  weigh  pounds. 

(c)  Sixty-five  bushels  of  wheat  weigh  how  much  more 
than  65  bushels  of  oats  ? 

11.  The  area  of  a  10-inch  square  is  ■  square  inches. 

(d)  How  many  square  feet  in  a  15-foot  square? 

(e)  How  many  square  yards  in  a  1 5-foot  square  ? 

12.  The  solid  content  of  a  10-inch  cube  is  cu.  in. 

(f)  How  many  cubic  feet  in  a  15-foot  cube? 

(g)  How  many  cubic  yards  in  a  15-foot  cube? 

13.  Three  fourths  of  60  are  .     60  is  f  of  . 

(h)  Find  I  of  132.         (i)  132  is  f  of  what  number? 

14.  Two  thirds  of  27  are  1  half  of  . 

(j)   Two  thirds  of  132  are  1  half  of  what  number? 

*  These  figures  refer  to  notes  on  page  8. 


30  COMPLETE    ARITHMETIC. 

MISCELLANEOUS   PROBLEMS. 

1.  Byron  bought  6  melons  for  54^ ;  he  sold  them  at  15^ 
each ;  on  each  melon  he  gained  cents.    ' 

(a)  A  merchant  bought  6  barrels  of  apples  for  $13.50, 
and  sold  them  at  $2.75  a  barrel.  How  much  did  he  gain  on 
each  barrel  ? 

2.  James  carried  4  dozen  eggs  to  market ;  he  sold  them 
for  12^  a  dozen ;  the  dealer  paid  James  for  the  eggs  with 

sugar  at  5^  a  lb. ;  James  should  receive  and ^ 

pounds  of  sugar. 

(b)  A  farmer  took  four  cords  of  wood  to  market ;  he  sold 
it  for  $6.75  a  cord  and  took  its  value  in  coal  at  $6  a  ton. 
How  many  tons  of  coal  should  he  receive  ? 

3.  At  5^  a  quart,  1  gallon  of  milk  is  worth  cents. 

2|-  gallons  are  worth  cents. 

(c)  At  5^  a  quart,  how  much  are  45|^  gallons  of  milk  worth  ? 

4.  Two  and  1  half  feet  are  inches.     4  weeks  are 

days. 

(d)  Forty-six  and  1  half  feet  are  how  many  inches  ? 

(e)  Fifty-two  weeks  are  how  many  days  ?* 

(f )  Seventy-five  gallons  are  how  many  quarts  ? 

5.  William  had  30^- ;  he  spent  |  of  his  money  for  ink  and 
pencils  and  the  remainder  for  paper ;  the  paper  cost cents. 

(g)  William's  father  had  $585 ;  he  spent  f  of  his  money 
for  a  carriage  and  liarness  and  the  remainder  for  horses. 
How  much  did  the  horses  cost  ? 

*  The  pupil  should  see  that  in  the  solution  of  this  problem  he  may  take  fifty-two 
7's  or  seven  52's.  If  he  adopts  the  latter  method  his  thought  may  be— (Me  day  in 
each  week  would  make  52  days,  and  seven  days  in  each  week  would  make  7  times  52 
days;  or  he  may  think  that  fifty-two  7's  are  equal  to  seven  52's,  and  that,  as  a  matter 
of  convenience,  he  finds  seven  52's. 


PART    I. 


31 


SIMPLE    NUMBERS. 

1.  A  number  that  has  no  exact  integral  divisors  except 
itself  and  1,  is  called  a  prime  number.  1,  2,  3,  5,  7,  11, 
,  ,  ,  ,  ,  are  prime  numbers. 

2.  Integral  numbers  that  are  not  prime  are  said  to  be 

composite.     4,  6,  8,  9.  ,  ,  ,  ,  are  composite 

numbers. 

3.  24  is  a  number.  23  is  a  - 

29  is  a  number.  26  is  a  - 

28  is  a  number.  27  is  a  - 

31  is  a  number.  37  is  a  - 


number, 
number, 
number, 
number. 


4.  If  an  integral  number  is  expressed  by  two  or  more 
figures,  and  the  right-hand  figure  is  5,  the  number  is  . 

5.  Every  even  number,  except  2,  is  . 

6.  Some  odd  numbers   are    ,  and  some  are    . 

9  is  .     11  is  .     13  is  .     21  is  . 

(a)   Write  all  the  prime  numbers  from  1  to  53  inclusive, 
and  find  their  sum. 


(b) 

(c) 

(d) 

(e) 

(f) 

Add. 

Subtract. 

Multiply. 

Divide. 

Divide. 

346 

5423 

754J 

S52)$7436* 

52)$7436t 

275 

1896| 

26 

142 

879 

(g) 

(h) 

(i) 

(J) 

27 

74624 

846 

53  bu.)2438  bu. 

53)2438  bu. 

624 

1827 

35|- 

*  This  means,  find  liow  many  times  $52  are  contained  in  $7436.  $52  are  contained 
in  $7436, 143  times.  Story— A  man  paid  $71,36  for  land  at  $52  an  acre;  there  were  lUS 
acres. 

t  This  means,  find  1  fifty-second  of  $7436.  1  fifty-second  of  $7436  is  $143.  Story— 
A  man  paid  $7it36  for  52  acres  of  land;  one  acre  cost  $1US. 


32  COMPLETE    ARITHMETIC. 

COMMON   FRACTIONS. 

Reduce  to  their  lowest  terms  : 

112_         12—  5—  8_  8_         io__ 

■*-Tcr—         3li  —         3J—        TiT—        T8—        TIT  — 

(a)T'A=  (b)TVtr=  (c)tVV  = 

Reduce  to  whole  or  mixed  numbers : 

2.  j/=    -v-=    -v-=    -V-=    -V-=    ¥=    -V-  = 

(d)i|i=  (e)H^=  (f)^«A  = 

Reduce  to  equivalent  fractions  having  a  common  denominator : 

3.  i,  i,  ^.      (Change  to  .)*  i  =  i  =  i  = 

(§)  i>  f'  I-  W  h  h  f-  w  i>  h  T- 

4.  Add  ^,  ^,  and  J.     (Change  to ths.) 

(j)  Find  the  sum  of  75|,  86|,  and  47f 

5.  From  -|-  subtract  i.     (Change  to  ths.) 

(k)  Find  the  difference  of  946|  and  SSSf 

6.  Divide  7  by  |.     This  means  .     Story.^ 

(1)  Find  the  quotient  of  36  divided  by  J.     Story. % 

(m)2^  gallons)! 2 6  gallons.  This  means,  ^nd  how  many 

'■  times  2\  gallons  are  qontained 

in  126  gal.     (Change  2^  and  126  to  fourths.)     Story. ^ 

(n)  4)276^  gallons.      This  means,  find  1  fourth  of  2161  gal- 

Ions.    Story — In  4  days  Mr.  Smith   sold 

276 jt  gallons  of  milk ;  this  was  at  the  rate  of  gallons  per  day. 

*  The  pupil  is  expected  to  find  by  trial  that  he  can  change  halves,  thirds,  and 
eighths,  to  twenty-fourths. 

t  See  note  (4)  page  6,  and  problem  4,  page  22. 

t  Mr.  Brown  put  36  bushels  of  peaches  into  J-bushel  baskets;  there  were  — 
baskets. 

§  Put  126  gal.  of  milk  into  2i-gal.  cans. 


PART    1.  33 

DECIMAL    FRACTIONS. 

1.  One  tenth  of  $2.45  is  .         .2  of  $2.45  = 

(a)  Multiply  $2.45  by  2.3.      This  means,  find  2  times 
$2.45  plus  3  tenths  of  $2.45. 

Operation.  Explanation. 

^2.45  One  tenth  of  $2.45  is  . 

2,3  Three  tenths  of  |2.45  are  -. 

Two  times  |2.45  are  . 

$.735  + 14.90  =  $5,635. 


$.735* 
$4.90 

$5,635 


NUMBER   STORY. 


If  1  ton  of  coal  is  worth  $2.45, 
1  tenth  of  a  ton  of  coal  is  worth 


3  tenths  of  a  ton  of  coal  are  worth  . 

2  tons  of  coal  are  worth  . 

2.3  tons  of  coal  are  worth  . 

(b)  Multiply  $3.65  by  2.4.*      (.1  of  $3.65  is  $.365.) 

(c)  Multiply  $52.8  by  3.2.        (.1  of  $52.8  is  $5.28.) 

2.  I  bought  7f  yards  of  print  at  6^  a  yd.  and  gave  the 
salesman  half  a  dollar;  I  should  receive  in  change  . 

(d)  I  bought  5.3  tons  of  coal  at  $4.20  a  ton  and  gave  the 
salesman  3  10-dollar  bills.  How  much  change  should  I 
receive  ? 

(e)  (f)               (g)  (h)  (i) 

Add.  Subtract.  Multiply.  Divide.  Divide. 

64f  78                3.75  $.5)$47.5t  5)$47.5j: 

28.37  24.42             2.6 

*  Write  the  decimal  point  in  each  partial  product  immediately  after  making  tfie 
figure  that  represents  the  tenths  in  that  product. 

t  This  means,  ^?id  how  many  times  5  tenths  are  contained  in  U75  tenths. 
%  This  means,  find  1  fifth  offU7.5. 


34 


COMPLETE    ARITHMETIC. 


DENOMINATE    NUMBERS. 


1.  One  ton  is 


pounds. 


2.  One  tenth  of  a  ton  is  — 

3.  Three  tenths  of  a  ton  are 

4.  Nine  tenths  of  a  ton  are 


pounds. 

—  pounds. 

—  pounds. 

pounds. 

6.  Three  hundredths  of  a  ton  are  pounds. 

7.  Seven  hundredths  of  a  ton  are  pounds. 


5.  One  hundredth  of  a  ton  is 


8.  One  thousandth  of  a  ton  is  — 

9.  Two  thousandths  of  a  ton  are 

10.  Six  thousandths  of  a  ton  are  ■ 

11.  2400  lb.  =  1  and  - 

12.  2800  lb.  =  1  and  - 

13.  3200  lb.  =  1  and  - 

14.  2460  lb.  =  1  and  - 

15.  2468  lb.  =  1  and  - 


pounds. 

—  pounds. 

-  pounds. 


tenths  (1.2)  tons, 
tenths  (  )  tons, 
tenths  (      )  tons. 

tons. 

tons. 


(a)  Change  4870  lb.  to  tons, 

(c)  Change  5260  lb.  to  tons, 

(e)  Change  6480  lb.  to  tons. 

16.  2  tons  are  lb. 

17.  2.3  tons  are  lb. 

18.  .03  of  a  ton  are  lb. 

19.  2.04  tons  are  lb. 

20.  2.34  tons  are  lb. 


(b)  4980  lb. 
(d)  5750  lb. 
(f)  7260  lb. 

•3  of  a  ton  are 
2.4  tons  are  — 


lb. 


lb. 


.04  of  a  ton  are  

2.03  tons  are  lb. 

3.23  tons  are  lb. 


lb 


(g)  Change  3.52  tons  to  lb.  (h)  4.37  tons, 

(i)    Change  5.18  tons  to  lb.  (j)  3.72  tons, 

(k)  Change  1.48  tons  to  lb.  (1)  2.324  tons, 

(m)  At  ^  a  cent  a  pound,  find  the  cost  of  3.45  tons  of 
scrap  iron. 


PART    I.  35 

MEASUREMENTS. 


A  /       An  \  An 

right  angle  /  acute  angle  \    obttise  angle 


1.  Wlien  two  lines  meet  at  a  point,  they  are  said  to 
form  an  angle. 

2.  When  a  jackknife  is  half  way  open,  the  handle  and 
blade  form  a  right  angle.     If  it  is  less  than  half  way  open, 

they  form  an  angle.     If  it  is   more  than  half  way 

open,  but  not  fully  open,  they  form  an  angle. 

3.  A  square  D  has  4  right  angles. 

4.  An  oblong  en  has  right  angles. 

5.  Rectangular  means  right-angled. 

6.  A  rectangular  figure,  or  rectangle,  may  be  a  square, 
or  it  may  be  an  ohlong. 

7.  A  rectangle  6  inches  by  6  inches  is  a  . 

8.  A  rectangle  4  in.  by  6  in.  is  an  . 

9.  A  rectangular  surface  4  feet  by  6  feet  is  an  ;  its 

area  is  square  feet  and  its  perimeter  is  feet. 

10.  A  rectangular  surface  8  feet  by  8  feet  is  a  ;  its 

area  is  square  feet ;  its  perimeter  is  ft. 

11.  Which  has  the  greater  area,  a  5-in.  square,  or  an 
oblong  4  in.  by  6  in.?     Compare  their  perimeters. 

Find  the  area  and  the  perimeter  of  each  of  the  following  rectan- 
gular surfaces : 

(a)  23  feet  by  6  feet.  (b)  25  inches  by  7  inches. 

(c)  15  yards  by  8  yards.  (d)  32  feet  by  23  feet, 

(e)  30  inches  by  20  inches.       (f)  21  yards  by  24  yards. 


36  COMPLETE    ARITHMETIC. 


RATIO    AND    PROPORTION. 

1.  One  sixth  of  12  is  .  12  is  -J  of 

2.  One  fifth  of  12  is  .  12  is  i  of 

3.  One  fourth  of  11  is  .       11  is  ^  of 


(a)  Find  1  third  of  275.         (b)  275  is  1  third  of  what? 
(c)  Find  1  half  of  377.  (d)  377  is  1  half  of  what  ? 

4.  Two  fifths  of  30  are  .      30  is  f  of  . 

5.  Three  fifths  of  30  are  .    30  is  f  of  . 

(e)  Find  f  of  90.  (f)  90  is  |  of  what  number? 

(g)  Find  f  of  90.  (h)  90  is  f  of  what  number? 

6.  14  is of  21.  21  is of  14. 

7.  21  is  . of  28.  28  is  • of  21. 

8.  28  is of  35.  35  is of  28. 

9.  Two  fifths  of  30  are  1  half  of  . 

10.  Three  fifths  of  30  are  2  thirds  of  . 


(i)  Two  fifths  of  90  are  1  half  of  what  number? 
(j)  Three  fifths  of  90  are  2  thirds  of  what  number? 

11.  Twelve  is   of  8,  or  and  ■ 

times  8.     A  man   can  earn  and times    as 

much  in  12  days  as  he  can  earn  in  8  days.     If  he  can  earn 
$20  in  8  days,  in  12  days  he  can  earn  . 

(k)  If  a  man  can  earn  $73  in  8  months,  how  many  dollars 
can  he  earn  in  12  months  ? 

12.  If  8  lb.  of  sugar  are  worth  50^,  12  lb.  are  worth  

cents. 

(1)  If  8  tons  of  coal  are  worth  $34.20,  how  much  are  12 
tons  worth  ? 

(m)  If  12  barrels  of  apples  are  worth  $27.60,  how  much 
are  8  barrels  worth  ? 


PART    I. 


37 


PERCENTAGE. 


11-J-  per  cent  = 
10    per  cent  = 

(1) 

1.  ll^per  cent  of  27  = 

2.  10    per  cent  of  20  = 

3.  11|  percent  of  18  = 

4.  10    per  cent  of  30  = 

5.  50    per  cent  of    3  = 

6.  25    per  cent  of    9  = 

7.  33i  per  cent  of    7  = 

8.  20    percent  of  11  = 

9.  16f  per  cent  of  19  = 

10.  141  per  cent  of  15  = 

(5) 

11.  G  is  per  cent  of  24. 

12.  6  is  per  cent  of  12. 

13.  3  is  per  cent  of  24. 

14.  3  is  per  cent  of  18. 


.10    = 


27 

20 

18 

30 

3 

9 

7 

11 

19 

15 


.1 

■7- 
1 

{2) 

\\\%  of  . 

10%  of  . 

11|%  of  . 

10%  of  . 

50%  of  . 

25%  of  . 

331-%  of  . 

20%  of  . 

16f  %  of  . 

142%  of  -^ — . 

(5) 

is  %  of  18. 

is  %  of  30. 

is  %  of  21. 

is  %  of  27. 


15.  Helen  bought  a  piece  of  flannel  that  was  40  inches 
long;  by  washing  it  shrank  10%  in  length;  after  washing 
it  was  inches  long. 

16.  Before  washing,  a  piece  of  flannel  was  40  inches  in 
length;  after  washing  it  was  35  inches  long;  it  shrank  by 
washing  per  cent* 

17.  A  dealer  had  25  bu.  of  apples;  he  lost  20%  of  them 
by  decay ;  there  remained  bushels. 

(a)  A  dealer  had  2375  bu.  of  apples;  he  lost  20%  of 
them  by  decay.     How  many  bushels  remained  ? 

*  It  shrank  what  part  of  its  original  length? 


'SS  COMPLETE    ARITHMETIC. 

PERCENTAGE. 

1.  lli%of36=  lli%of38  = 

(a)  Find  11^  per  cent  of  1044;  (b)of  1047. 

2.10^0  of  50=  10%  of  51=  10%  of  52  = 

(c)  Find  10%  of  870  ;     (d)  of  874. 

3.  7  is  11-1-%  of  .  7|  is  11^%  of  . 

(e)  371  is  lli%  of  what?      (f)  371i  is  lli-%  of  what  ? 

4.  8  is  10%  of  .  81-  is  10%  of  . 

(g)  89  is  10%  of  what  ?  (h)  89.2  is  10%  of  what  ? 

(3) 

12|-  is  %  of  50. 

6  is  %  of  54. 

3J  is  %  of  10. 


5.  12  is  ■ 

%  of  24. 

6.  6  is  - 

— %  of  60. 

7.  3|-  is  - 

%  of  7. 

8.  36  is  ■ 

%  of  72. 

(1) 

9.  One  per  cent  of  $700  is 

10.  One  per  cent  of  $730  is 

11.  One  per  cent  of  $732  is 

(i)    Find  3 

%  of  $732. 

(k)  Find  5 

%  of  $732. 

(m)  Find  7 

%  of  $320. 

72  is  %  of  144. 


2%  of  $700  = 
2%  of  $730  = 
2%  of  $732  = 


(j)  Find  4%  of  $732 
(1)  Find  6%  of  $732. 
(n)  Find  7%  of  $326. 

12.  Two  dollars  are  1  hundredth  of  $200. 

13.  Two  dollars  are  1%  of  .  $4  are  2%  of 

14.  Six  dollars  are  3%  of  .  $8  are  4%  of 

15.  Five  dollars  are  1%  of  .         $10  are  2%  of 


PART    I.  39 


REVIEW. 


1.  All  integral  numbers  are  either  prime  or  composite. 
17  is  .     31  is  .     95  is  .      242  is  .     370 


IS 


2.  Keduce  each  of  the  following  improper  fractions  to  a 
whole  number  or  to  a  mixed  number :  -y-,  ^^-,  -y-,  ^^-,  \K 

(a)  ^±.     (b)  ^K     (c)  ^5  6.     (d)  ^K 

3.  Eeduce  each  of  the  following  to  its  lowest  terms : 

8  9  12  15  11  12  10 

^0"  ^T  ITT  ¥6-  3T  T¥  TUir 

(e)TVV         (OtVV         (g)Hf         WtW 

4.  Eeduce  the  following  to  equivalent  fractions  having 
a  common  denominator: 

i>if      (Change  to  .)      ^=  i=  ^  = 

(i)  h  I  h  (3)  h  h  h  W  I'  h  h 

5.  If  ^  of  a  bushel  of  potatoes  costs  20^,  2|-  bushels 
will  cost  . 


(1)  If  ^  of  a  ton  of  straw  costs  $2.25,  how  much  will  3.2 
tons  cost? 

6.  Change  the  following  to  tons :  4200  lb. 
(m)  5750  lb.         (n)  7320  lb.         (o)  3150  lb. 

7.  Find  the  area  and  the  perimeter  of  each  of  the  fol- 
lowing rectangular  surfaces :  7  feet  by  5  feet. 

(p)  27  feet  by  12  feet.         (q)  15  inches  by  15  inches. 

8.  If  2  gallons  of  molasses  are  worth  6 Of/,  3  gallons  are 
worth  cents. 

(r)  If  2  loads  of  brick  are  worth  $12.60,  how  much  are 
3  loads  worth  ? 


40  COMPLETE   ARITHMETIC. 

MISCELLANEOUS    PROBLEMS. 

1.  In  a  school  there  are  35  pupils ;  ^  of  the  pupils  are 

boys ; of  the  pupils  are  girls ;  there  are  boys 

and  girls. 

(a)  In  a  school  there  are  392  pupils ;  |  of  the  pupils  are 
boys.     How  many  girls  are  in  the  school  ? 

2.  If  3  melons  cost  36^,  at  the  same  rate  5  melons  will 
cost  cents. 

(b)  If  3  acres  of  land  cost  $525,  how  much  will  5  acres 
cost  at  the  same  rate  ? 

3.  Harry  exchanged  5  lb.  of  butter  at  20^  a  pound  for 

coffee  at  25^  a  pound;  he  should  receive  pounds  of 

coffee. 

(c)  Harry's  father  exchanged  6  cords  of  wood  at  $5.20  a 
cord  for  cedar  posts  at  20^  each.  How  many  posts  should 
he  receive  ? 

4.  3  +  2  +  6+4  +  5  +  8+1  +  7  +  9  +  4  +  7  +  3  +  2  = 

(d)  275  +  361  +  554  +  732  +  598  +  236  +  347  +  256  = 

5.  If  Mark  saves  $4  a  month,  in  1  year  he  will  save 

dollars. 

(e)  If  Mark's  father  saves  $21.50  a  month,  how  much 
will  he  save  in  1  year  ? 

6.  A  common  brick  is  8  inches  long,  4  in.  wide,  and  2  in. 
thick ;  it  has  two  faces  each  of  which  is  4  in.  by  8  in.,  two 

faces  each  of  which  is  «  in.  by  in.,  and  two  faces 

each  of  which  is  in.  by  in. 

(f)  Find  the  sum  of  the  areas  of  all  the  faces  of  a  common 
brick. 

(g)  Change  674  inches  to  feet  and  inches. 


PART    I.  41 

SIMPLE    NUMBERS. 

1.  Any  exact  integral  divisor  of  a  number  (except  the 
number  itself  and  1)  is  called  a  factor  of  the  number.     The 

factors  of  6  are  and  .     The  factors  of  10  are  

and  . 

2.  A  factor  that  is  itself  a  prime  number  is  called  a 
prime  factor.  A  factor  that  is  itself  a  composite  number  is 
called  a  composite  factor. 

2  and  3  are  factors  of  24. 

4  and  6  are  factors  of  24. 

3.  Every  composite  number  may  be  resolved  into  prime 
factors. 

The  prime  factors  of  12  are  2,  2,  and  3. 
The  prime  factors  of  18  are  2,  3,  and  3. 

The  prime  factors  of  15  are  •  and  . 

The  prime  factors  of  14  are and  . 

The  prime  factors  of  30  are  , ,  and  . 

4.  From  the  above  it  will  be  seen  that  a  number  is 
equal  to  the  product  of  its  prime  factors.  3  and  7  are  the 
prime  factors  of  ;  2,  2,  and  7,  of  . 

2)30        The  prime  factors  of      2)50        The  prime  factors  of 
3)15        30  are  2,  3,  and  5.  5)25        50  are  2,  5,  and  5. 

■      5        2  X  3  X  5  =  30.  5        2  X  5  X  5  =  50. 

(a)  What  are  the  prime  factors  of  40  ?         (b)  Of  60  ? 
(c)   Of  65  ?        (d)  Of  72  ?        (e)  Of  86  ?        (f)  Of  85  ? 

Multiply.  Divide.  Divide. 

(g)  724  by  28.  (h)  748  lb.  by  32  lb.  (i)  1254  lb.  by  9. 

(j)    846  by  23.  (k)  834  lb.  by  32  lb.  (1)  1046  lb.  by  9 

(m)  637  by  25.  (n)  928  lb.  by  32  lb.  (o)  1134  lb.  by  9. 

(p)  926  by  27.  (q)  796  lb.  by  32  lb.  (r)  1 341  lb.  by  9. 


39  _ 


42  COMPLETE   ARITHMETIC. 

COMMON   FRACTIONS. 

Reduce  to  improper  fractions  : 

1.  7i  =  y.         8|  =  ^.         9|  =  ^.         7|=,. 

(a)  27^=  (b)48f=         (c)75f  = 

Reduce  to  whole  or  mixed  numbers : 

Ol7_  35—  39—  38—  45-- 

^'    --6-  -  S-  -  "¥"  -  ~3 "  -  ~V  - 

(d)2  8A=  (e)^^^  (^)H^  = 

Reduce  to  equivalent  fractions  having  a  common  denominator : 

3.  i,  i,  i.      (Change  to  .)  ^=  ^  =  i  = 

(g)  h  h  T-        W  -6'  T'  I-        ©  h  h  h 

4.  Add  i,  ^,  and  ^.     (Change  to  .) 

(j)  Find  the  sum  of  86|,  95|,  and  87f 

5.  From  |  subtract  ^.     (Change  to  .) 

(k)  Find  the  difference  of  873f  and  249|. 

6.  Divide  |-  by  -i.     This  unesins,  find  how  many  times  ^ 
is  contaiiud   in    |.     I    can    change   fourths   and  thirds   to 

ths.     I  =  y^.     ^  =  -YY-     twelfths  are  contained  in 

twelfths  times.     Story — James  can  husk  a  row 

of  corn  in  i  of  an  hour ;  in  f  of  an  hour  he  can  husk  

and rows. 

(l)i-^i.  (m)l-i.  (n)f-f 

(o)  5-1-  acres)288  acres.       This  means,  find  how  many  times 

5^  acres  are  contained  in  288  acres. 

(p)  5  )6 2  5 1^  acres.     This  means,  find  1  fifth  of  625\  acres.  Story. 


PART    I.  43 


DECIMAL    FRACTIONS. 


1.  One  hundredth  of  $500  is  .         .02  of  $500  = 

2.  One  hundredth  of  $540  is  .         .02  of  $540  - 

3.  One  hundredth  of  $542  is  -.         .02  of  $542  -^ 


4.  .01  of  $600=  .01  of  $60=  .01  of 

5.  .01  of  $.1  =  .01  of  $.5  =  .01  of  $.7  = 

6.  .01  of  $6.4  =  .01  of  $7.5  =  .01  of  $3.2  = 

7.  .01  of  $24.2  =  .01  of  $37.1  =  .01  of  $53.1  = 

(a)    Multiply  $374  by  .03.    This  means,  ^Tif?  3  hundredths 
of  $374. 

Operation.  Explanation. 


$374 


* 


One  hundredth  of  $374  is  

.03  Three  hundredths  of  ^374  are 


$11.22 

NUMBER   STORY. 
If  one  acre  of  land  is  worth  f  374, 

1  hundredth  of  an  acre  of  land  is  worth . 

3  hundredths  of  an  acre  of  land  are  worth  


(b)  Multiply  $347  by  .03.         (c)  $537  x  .04. 
(d)  Multiply  $24.6  by  .03.        (e)  $39.4  x  .04. 
(f)  At  $875  an  acre,  how  much  will  .04  of  an  acre  of 
land  cost  ? 

(g)              (h)              (i)  (j)                       (k) 

Add.  Subtract.  Multiply.  Divide.  Divide. 

286.3           146f            356  $.04)$5.76t  4)$5.76| 
184f              78.2            .05 

*The  pupil  should  understand  that  he  multiplies  $3.74  (not  §371)  by  3,  and 
should  be  taught  to  write  the  decimal  point  in  the  product  immediately  after  writing 
the  tenths'  figure  of  the  product. 

fThis  means, ^nd  how  many  times  U  hundredths  are  contained  in  576  hundredths. 

X  This  means,  ^?id  1  fourth  of  $5.76. 


44 


COMPLETE    ARITHMETIC. 


DENOMINATE    NUMBERS. 


1.  4000  lb.  are 

2.  4200  lb.  are 

3.  4400  lb.  are 

4.  4600  lb.  are 


tons, 
tons, 
tons, 
tons. 


(a)  At  $5.20  a  ton,  how  much  will  4600  lb.  of  coal  cost? 


Operation. 
4600  lb.  =  2.3  tons. 

$5.20 
2.3 


$1,560 
$10.40 
$11,960 


Explanation. 
One  ton  costs  ^5.20. 

1  tenth  of  a  ton  costs  $.52. 
3  tenths  of  a  ton  cost  

2  tons  cost  . 

2.3  tons  cost  . 


Find  the  cost : 
(b)  4800  lb.  @  $5.40  per  ton. 
(d)  6200  lb.  @  $5.35  per  ton. 
(f)  5600  lb.  @  $5.70  per  ton. 

5.  Three  yd.  are feet. 

6.  Three  hr.  are minutes. 

7.  Three  min.  are seconds. 

8.  Three  bu.  are quarts. 

9.  Three  lb.  are ounces. 


(c)  4200  lb.  @  $5.60. 
(e)  6400  lb.  @  $5.25. 
(g)  5400  lb.  @  $5.80. 

3  yd.  are inches. 

3  yr.  are months. 

3  wk.  are days. 

3  gal.  are quarts. 

3  pk.  are pints. 


(h)  Change  148  yd.  to  feet,   (i)   Change  15  yd.  to  inches. 


(j)  Change  24  hr.  to  min. 

(1)  Change  35  yr.  to  mo. 

(n)  Change  37  lb.  to  oz. 

(p)  Change  43  bu.  to  qts. 


(k)  Change  24  hr.  to  seconds, 
(m)  Change  52  wk.  to  days, 
(o)  Change  49  gal.  to  quarts, 
(q)  Change  56  pk.  to  pints. 


(r)    How  many  square  inches  in  a  5 -foot  square 


PART    I. 
MEASUREMENTS — RECTANGULAR   SOLID 


46 


1.  A  rectangular  solid  has  faces.     Each  face  is  a 

rectangle. 

2.  Some  or  all  of  the  faces  of  a  rectangular  solid  may  be 
squares. 

3.  If  each  face  of  a  rectangular  solid  is  a  square,  the 
solid  is  called  a  . 

4.  If  some  of  the  faces  of  a  rectangular  soHd  are  oblongs, 
the  sohd  is  not  a  cube. 

5.  The  area  of  a  rectangular  surface  3  inches  by  4  inches 
is  square  inches. 

6.  The  solid  content  of  a  rectangular  solid  3  in.  by  4  in. 
by  2  in.  is  cubic  inches. 

7.  The  area  of  a  rectangular  surface  3  inches  by  5  inches 
is  square  inches. 

8.  The  solid  content  of  a  rectangular  solid  3  in.  by  5  in. 
by  2  in.  is  cubic  inches. 

9.  The  area  of  a  rectangular  surface  2  inches  by  5  inches 
is  square  inches. 

10.  The  solid  content  of  a  rectangular  solid  2  in.  by  5  in. 
by  3  in.  is  cubic  inches. 


46  COMPLETE    ARITHMETIC. 


RATIO   AND    PROPORTION. 

1.  One  seventh  of  28  is  .  28  is  i  of  . 

2.  One  fifth  of  28  is  .  28  is  ^  of  . 

3.  One  third  of  28  is  .  28  is  i  of  . 

(a)  Find  1  fourth  of  387.    (b)  387  is  1  fourth  of  what  ? 
(c)  Find  1  fifth  of  724.       (d)  724  is  1  fifth  of  what  ? 

4.  Five  sixths  of  60  are  .  60  is  |  of  . 


5.  Four  fifths  of  40  are  .  40  is  |  of  . 

(e)  Find  |  of  420.  (f)  420  is  |  of  what  number  ? 

(g)  Find  4  of  920.  (h)  920  is  ^  of  what  number  ? 

6.  16  is ot  24.  24  is of  16. 

7.  24  is of  32.  32  is of  24. 

8.  32  is of  40.  40  is of  32. 

9.  40  is of  48.  48  is of  40. 

10.  Five  sixths  of  30  are  1  third  of  . 

11.  One  sixth  of  48  is  2  thirds  of  . 


(i)  Five  sixths  of  366  are  1  half  of  what  number? 
(j)  One  sixth  of  852  is  2  thirds  of  what  number? 

12.  Twelve  is of  9,  or   and   


times   9.     A  man  can  earn  and times  as 

much  in  12  days  as  he  can  earn  in  9  days.     If  he  can  earn 

$24  in  9  days,  in  12  days  he  can  earn  dollars. 

(k)  If  a  man  can  earn  $840  in  9  months,  how  many  dol- 
lars can  he  earn  in  12  months. 

13.  If  9  lb.  of  nails  are  worth  33  cents,  12  lb.  are  worth 

cents. 

(1)  If  9  cords  of  wood  are  worth  $42.75,  how  much  are 
12  cords  worth  ? 


PART    I. 


47 


PERCENTAGE. 


66|  per  cent  =  .66|  =  |. 
75  per  cent  =  .75  =:  -|. 


1.  66f  per  cent  of  12  = 

2.  75  per  cent  of  12  = 

3.  66f  per  cent  of  24  = 

4.  75  per  cent  of  24  = 

5.  50  per  cent  of  5  = 

6.  25  per  cent  of  13  = 

7.  33i  per  cent  of  10  = 

8.  20  per  cent  of  16  = 

9.  16f  per  cent  of  25  = 

10.  14f  per  cent  of  22  = 

11.  12-1-  per  cent  of  33  = 

12.  11|  per  cent  of  19  = 

13.  10  per  cent  of  21  = 


(5) 

12  is  66f  %  of 

12  is  75%  of  - 

24  is  66|%  of 

24  is  75%  of  - 

5  is  50%  of  - 

13 

10 

16 

25 

22 

33 

19 

21 


14.  4  is 

15.  4  is 

16.  4  is 


(3) 

—  per  cent  of  24. 

—  per  cent  of  28. 

—  per  cent  of  32. 

—  per  cent  of  36. 
per  cent  of  40. 


17.  4  is 

18.  4  is 

19.  9  is  per  cent  of  12. 


s  25%  of  - 
s  33^%  of 
s  20%  of  - 
s  16f  %  of 
s  14f  %  of 
s  121-%  of 
s  lli%  of 


s  10%  of 


7  is 
7  is 

7  is 

7  is 

8  is 
12  is 


— %  of  35. 
— %  of  28. 
— %  of  21. 

— %  of  14. 
-%  of  12. 

%  of  18. 


20.  Twenty-five  per  cent  of  80  sheep  are  sheep. 

21.  Twenty-one  sheep  are  25%  of  sheep. 

22.  Twenty-five  sheep  are  per  cent  of  75  sheep. 

23.  Forty  sheep  are  per  cent  of  60  sheep. 


48  COMPLETE   AKITHMETIC. 

PEKCENTAGE. 

1.  75  per  cent  of  36  =  66f  per  cent  of  36  = 

(a)  Find  75%  of  796.  (b)  Find  66f  %  of  822. 

(c)  Find  3%  of  $375.     This  means,  ^tic?  3  hundredths  of 
375. 

Operation.  Explanation. 

$375  One  per  cent  of  |375  =  $3.75. 

Q3  Three  per  cent  of  $375  =  $11.25. 


$11.25 


NUMBER   STORY. 


Mr.  A  collected  money  for  Mr.  B.  It  was  agreed  that  Mr.  A 
should  keep  S%  of  all  he  might  collect  to  pay  him  for  his  trouble. 
He  collected  $375 ;  he  should  keep  $11.25  and  "pay  over  "the  re- 
mainder to  Mr.  B. 

(d)  How  much  should  Mr.  A  "  pay  over  "  to  Mr.  B  ? 

(e)  Find  7%  of  $465.  (f)  Find  9%  of  $324. 
(g)  Find  3%  of  $422.            (h)  Find  7%  of  $538. 

2.  36  is  75%  of  .  36  is  66|%  of  . 

(i)  453  is  75%  of  what?      (j)  562  is  66|%  of  what? 

(k)  Forty-eight  dollars  are  3  %  of  what  ?  This  means, 
$48  are  3  hundredths  of  how  many  dollars  ? 

Operation.                                      Explanation. 
$48  -f-  3  =  $16.               C)ne  hundredth  of  the  unknown  number  is 
$16  X  100  =  $1600      ^^^'   ^^^  hundredths  (the  whole)  are 

NUMBER   STORY. 

A  lawyer  collected  some  money  for  3%  of  the  amount  collected ; 
his  share  (commission)  was  $48  ;  the  amount  collected  was  $1600. 

(1)  How  much  should  the  lawyer  pay  over  to  the  man  for 
w^hom  he  collected  the  money  ? 


PART   1.  49 

REVIEW. 

1.  The  prime  factors  of  45  are  ,  ,  and  . 

(a)  What  are  the  prime  factors  of  100  ?  (b)  Of  125  ? 

2.  Two,  3,  and  5  are  the  prime  factors  of  . 

(c)  Of  what  number  are  3,  3,2,  and  7  the  prime  factors  ? 

3.  Keduce  ^f  to  its  lowest  terms.         ^  = 

(d)  Reduce  ^^  and  ^-^  to  their  lowest  terms. 

4.  Reduce  8|-  to  an  improper  fraction.         8|-  = 

(e)  Reduce  57|  and  72|-  to  improper  fractions. 

5.  Reduce  -\^-  to  a  mixed  number.         -\8-  = 

(f)  Reduce  ^^  and  ^^  to  mixed  numbers. 

6.  Reduce  f  and  f  to  equivalent  fractions  having  a 
common  denominator.         |-  =         |-  = 

(g)  Reduce  y^  and  ^  to  equivalent  fractions  having  a 
common  denominator. 

7.  Multiply  60  by  .7.     This  means,  find  7  tenths  of  60. 
One  tenth  of  60  =  ;  7  tenths  of  60  =  . 

(h)  Multiply  $537  by  .07.    This  means,  ^ti^  7  hundredths 
of  $537.      (See  page  43.) 

8.  At  $1  per  ton,  4240  lb.  of  coal  cost  . 

(i)  Find  the  cost  of  4240  lb.  of  coal  at  $7  per  ton. 

9.  The  volume  of  a  rectangular  solid  5   inches  by  4 
inches  by  2  inches  is  cubic  inches. 

(j)  Find  the  volume  of  a  rectangular  soUd  9  inches  by  7 
inches  by  7  inches. 

10.  If  9  lb.  of  tea  are  worth  $6,  12  lb.  are  worth  . 

(k)  If  9  acres  of  land  are  worth  $346.50,  how  much  are 
12  acres  worth  at  the  same  rate  ?     (12  is  1^  times  9.) 


50  COMPLETE    ARITHMETIC. 

MISCELLANEOUS   PROBLEMS. 

1.  From  June,  1881,  to  June,  1899,  it  is  years. 

(a)  How  many  years  from  Aug.,  1492,  to  Aug.,  1897  ? 

2.  A  man  sold  a  horse  for  one  hundred  twenty  dollars ; 
this  was  seventeen  dollars  and  twenty  cents  more  than  the 
horse  cost  him  ;  the  horse  cost  him  . 

(b)  A  man  sold  a  farm  for  fourteen  thousand  seven  hun-- 
dred  fifty  dollars  ;  this  was  eight  hundred  seventy-five  dollars 
more  than  he  paid  for  the  farm.     How  much  did  the  farm 
cost  him  ? 

3.  From  5460  lb.  of  coal  there  were  sold  2  tons.     

pounds  remained. 

(c)  From  18940  lb.  of  coal  there  were  sold  8|-  tons. 
How  many  pounds  were  left  ? 

4.  From  Mendota  to  Galesburg  it  is  80  miles ;  a  train 
going  30  miles  an  hour,  that  leaves  Mendota  at  8:30,  should 
arrive  at  Galesburg  at  . 

(d)  From  Chicago  to  Denver  it  is  about  1000  miles.  If 
a  train  leaves  Chicago  for  Denver  at  9  o'clock  Monday 
morning  and  goes  at  the  rate  of  30  miles  an  hour,  when 
will  it  arrive  at  Denver  ? 

5.  If  butter  is  25^  a  pound  and  coffee  is  30^  a  pound 
3  lb.  butter  will  pay  for  lb.  coffee. 

(e)  Fifteen  and  one  half  pounds  of  butter  at  24^  a  pound, 
will  pay  for  how  many  pounds  of  coffee  at  32^  a  pound? 

6.  If  f  of  a  yard  of  lace  is  worth  15^,  2^  yards  are  worth 
cents. 

(f)  If  I  of  a  yard  of  cloth  costs  $1.05,  how  much  will 
27^-  yards  cost  ? 


PART    I.  51 

SIMPLE    NUMBERS.* 

1.  Four,  6,  8,  10,  12,  etc.,  are  multiples  of  2. 

2.  Six,  9,  12,  15,  18,  etc.,  are  multiples  of  3. 

3.  Twelve  is  a  multiple  of  2.     12  is  also  a  multiple  of 
3,  and  of  4,  and  of  6. 

4.  Twelve  is  a  common  multiple  of  2,  3,  ,  and  . 

5.  Fifteen  is  a  common  multiple  of  and  . 

6.  Common  multiples  of  4  and  6  are,  12,  24,  ,  etc. 

The  least  common  multiple  of  4  and  6  is  12. 

7.  Common  multiples  of  6  and  8  are,  24,  48,  ,  eta 

The  least  common  multiple  of  6  and  8  is  . 

8.  Common  multiples  of  8  and  12  are,  ,  ,  etc. 

The  least  common  multiple  of  8  and  12  is  . 

9.  Common  multiples  of  6  and  9  are, ,  ,  etc. 

The  least  common  multiple  of  6  and  9  is  . 

10.  The  prime  factors  of  18  are  ,  ,  and  . 

11.  The  prime  factors  of  70  are  ,  ,  and  . 

(a)  What  are  the  prime  factors  of  140.       (b)  Of  160  ? 
(c)  Of  135  ?      (d)  Of  175  ?      (e)  Of  250  ?      (f)  Of  225  ? 

Multiply.  Divide.  Divide, 

(g)    635  by  53.  (h)  944  lb.  by  56  Ib.f-  (i)  1536  lb.  by  12.t 

(j)    728  by  54.  (k)  846  lb. by  56  lb.    (1)  1445  lb.  by  12. 

(m)  834  by  52.  (n)  739  lb.  by  56  lb.    (o)  1374  lb.  by  12. 

(p)   947  by  51.  (q)  873  lb.  by  56  lb.    (r)  1653  lb.  by  12. 

(s)    836  by  55.  (t)  965  lb.  by  56  lb.    (u)  1738  lb.  by  12. 

*  Do  much  oral  work  in  preparation  for  this  page.  By  using  these  terms,  make 
the  pupil  as  familiar  with  multiple,  common  multiple,  and  least  common  multiple,  as  he 
is  with  house,  schoolhouse,  and  stone  schoolhouse. 

t  Tell  the  meaning.    Tell  a  number  story. 


52      ^  COMPLETE    ARITHMETIC. 

COMMON    FRACTIONS. 

L.  c.  m,  is  the  abbreviation  for  least  common  multiple. 


1. 

Add  1  and  \^. 

(a)  1+1- 

L.  c.  m.  of  8  and  12  is  . 

(b)  f +  i 

i  =  -ST-            TJ  =  Ti' 

(c)   l+f- 

iT  +  li  =  ^T=l¥¥- 

(d)  l  +  f 

2. 

Fromf  subtract  \. 

(e)  l-i- 

L.  c.  m.  of  9  and  6  is  . 

(f)  1-^- 

i      T¥-          "B  -  T8-- 

(g)TV-l- 

'ff-A-TT. 

(h)f-|. 

3. 

Divide  |J  by  f 

(i)   l^i- 

L.  c.  m.  of  12  and  8  is  . 

(i)  f^-J- 

■H  =  ^T-          i  =  IT- 

(k)f-i 

lf-A  = 

(1)  4^f 

4. 

Multiply  12  by  2f . 

(m)   48x2t. 

This  means  . 

(n)  252  X  3f 

23  times  12  -         .     Story. 

(0)  175  X  2|. 

5. 

Multiply  12  by  |. 

(P)  96  xf. 

TVii"  moTn" 

(q)  84  X  |. 

1  of  12  =  .     Story. 

(r)  95  X  f. 

(s)  6^  dollars)375  dollars.     Thi&mQd^ns,  findhowmany  times 

$6\  are  contained  in  $375. 
(Change  6^  and  375  to  fourths.)    Story, 

(t)  6)756  dollars.   This  means,  find  1  sixth  of  $156.    Story. 


PART   I.  53 

DECIMAL    FRACTIONS. 

1.  One  tenth  of  $432  is  .  .2  of  $432  = 

2.  One  hundredth  of  $432  is  .       .02  of  $432  := 

(a)  Multiply  $432  by  .25.     This  means,  find  2  tenths  of 
$432,  plus  5  hundredths  of  $432. 

Operation.  Explanation. 

$432  One  hundredth  of  $432  is  . 

.25  5  hundredths  of  $432  are  . 

One  tenth  of  $432  is  . 

2  tenths  of  $432  are  . 

$21.60 +  $86.4  = 


$21.60* 
$86.4  t 


$108.00 


NUMBER   STORY. 
If  1  acre  of  land  is  worth  $432, 
1  hundredth  of  an  acre  is  worth 


5  hundredths  of  an  acre  are  worth 

1  tenth  of  an  acre  is  worth  . 

2  tenths  of  an  acre  are  worth  


25  hundredths  of  an  acre  are  worth 


(b)  Multiply  $325  by  .23.  (c)  $482  x  .32. 

(d)  Multiply  $278  by  .43.  (e)  $356  x  .36. 

(f)  Multiply  $536  by  .07.  (g)  $351  x  .7. 

(h)  Multiply  $249  by  2.6.  (i)  $426  x  .45. 

(J)  (k)             (1)                 (m)  (n) 

Add.  Subtract.  Multiply.  Divide.  Divide. 

8.75  56-J-            675  $.05)$6.  5)$6. 

7.324  12.9            .36  ~" 

*  The  pupil  should  understand  that  he  multiplies  $4.32  (not  $432)  by  5,  and 
should  be  taught  to  write  the  decimal  point  in  the  partial  product  immediately  after 
writing  the  tenths'  figure,  6,  of  the  partial  product. 

t  The  pupil  should  understand  that  he  multiplies  $43.2  (not  $432)  by  2,  and 
should  be  taught  to  write  the  decimal  point  in  the  partial  product  immediately  after 
writing  the  tenths'  figure,  4,  of  the  partial  product. 


54  COMPLETE    ARITHMETIC. 

DENOMINATE    NUMBERS. 

1.  Sixteen  and  one  half  feet  are  1  rod. 

2.  Three  hitindred  twenty  rods  are  1  mile. 

3.  One  rod  is  and yards.* 

4.  Two  rods  are  feet.  4  rods  are  feet. 

5.  Six  rods  are  feet.  10  rods  are  feet. 

6.  One  mile  is  rods.  J  mile  is  rods. 

7.  \  mile  is  rods.  -i-  mile  is  rods. 

8.  The  telegraph  poles  along  the  line  of  a  railroad  are 

usually  ten  rods  apart ;  they  are  feet  apart.    From  the 

first  telegraph  pole  to  the  third  it  is  rods ;  from  the 

first  to  the  fifth  it  is  rods. 

(a)  How  far  is  it  from  the  first  telegraph  pole  to  the 
thirty-third  ? 

9.  The  roads  in  the  country  are  usually  4  rods  wide.    A 
4-rod  road  is  feet  wide. 

10.  A  100-foot  street  is  rods  and  foot  wide. 

11.  From  the  schoolhouse  to  ,  it  is of  a 

mile,  or  rods. 

(b)  Change  5  mi.  to  rods.         (c)  Change  40  rd.  to  feet. 
(d)  Change  28  rd.  to  yards,     (e)  Change  7  mi.  to  rods. 

REVIEW. 

12.  When  hay  is  $12  a  ton,  3000  lb.  cost  . 

13.  When  oats  are  30^  a  bushel,  96  lb.  cost  . 

14.  When  wheat  is  80^  a  bushel,  180  lb.  cost  . 

(f)  When  com  (not  shelled)  is  45^'  a  bushel,  how  much 
will  3640  lb.  cost  ? 

(g)  When  coal  is  $5.40  a  ton,  how  much  will  2600  lb.  cost  ? 

*  Discover  by  actual  measurement  the  number  of  yards  in  a  rod. 


PART   1. 

MEASUREMENTS. 


55 


1.  A  pile  of  vjood  8  feet  long,  4  feet  wide,  and  4  feet  high 
(or  its  equivale7it)  is  called  a  cord. 

(a)  How  many  cubic  feet  in  a  rectangular  solid  8  feet  by 
4  feet  by  4  feet  ? 

2.  A  pile  of  wood  16  feet  long,  4  feet  wide,  and  4  feet 
high  contains  cords. 

(b)  How  many  cubic  feet  in  2  cords  ? 

3.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  6  feet  high 
contains  cords.  • 

(c)  How  many  cubic  feet  in  1|-  cords  ?  (d)  In  f  of  a  cord  ? 
(e)  How  many  cubic  feet  in  5  ^-  cords  ?     (f)  In  6  J  cords  ? 


REVIEW. 


4.  A  1-inch  square  = 

5.  A  1-inch  cube  =  — 


6.  A  2 -inch  square  = 

7.  A  2-inch  cube  =  — 


-  of  a  2 -inch  square, 
of  a  2 -inch  cube. 

-  of  a  3-inch  square, 
of  a  3 -inch  cube. 


(g)  Find  the  area  of  a  surface  12  ft.  by  24  ft.* 

(h)  Find  the  volume  of  a  solid  12  ft.  by  8  ft.  by  4  ft.f 

*Take  care  that  the  pupil  understands  that  he  does  not  (cannot)  multiply  24  ft. 
by  12  ft.,  but  24  square  feet  by  12,  or  12  square  feet  by  24. 

fin  this  problem  the  pupil  multiplies  12  cubic  feet  by  8,  and  the  product  thus 
obtained  by  4. 


56  COMPLETE    ARITHMETIC. 


RATIO    AXD    PROPORTION. 

1.  One  eighth  of  24  is  .         24  is  |  of  . 

2.  One  seventh  of  24  is  .       24  is  ^  of  . 

3.  One  fifth  of  24  is  .  24  is  i  of  . 

(a)  Find  l  of  49.26.         (b)  49.26  is  J  of  what? 
(c)  Find  i  of  $9.31.         (d)  $9.31  is  i  of  what? 

4.  Four  sevenths  of  56  are  .  56  is  f  of  

5.  Three  sevenths  of  42  are  .        42  is  f  of  

(e)  Find  -f  of  875.         (f)  876  is  f  of  what  number? 
(g)  Find  I  of  924.         (h)  927  is  f  of  what  number? 

6.  18  is  of  27.         27  is of  18. 

7.  27  is of  36.         36  is of  27. 

8.  45  is of  54.         54  is of  45. 

9.  Three  sevenths  of  28  are  2  thirds  of  . 

10.  Six  sevenths  of  28  are  3  fourths  of  . 


(i)    Three  sevenths  of  364  are  1  half  of  what  number? 
(j)    One  seventh  of  434  is  2  thirds  of  what  number  ? 

11.  Twenty  is of  25.     If  25  bags  of  salt  are 

worth  $15,  20  bags  of  salt  are  worth  dollars. 

(k)  If  25  acres  of  land  are  worth  $640,  how  much  are 
20  acres  worth  at  the  same  rate  ? 

12.  Twenty-five  is  of   20,  or  and  


times  20.     If  20  bushels  of  apples  are  worth  $12, 

25  bushels  of  apples  are  worth  dollars. 

(1)    If  20  barrels  of  salt  are  worth  $22.40,  how  much  are 
25  barrels  of  salt  worth  at  the  same  rate  ? 

13.  If  12  qt.  of  nuts  are  worth  40^-,  9  qt.  of  nuts  are 
worth  cents. 


PART  I.  57 

PERCENTAGE. 

40  per  cent  =  .40  =  |. 
60  per  cent  =  .60  =  -|. 

{1)  (2) 

1.  40  per  cent  of  50  =  50  is  40%  of  

2.  60  per  cent  of  15  =  15  is  60%  of  — 

3.  40  per  cent  of  45  =  16  is  40%  of  — 

4.  60  per  cent  of  45  =  18  is  60%  of  — 

5.  75  per  cent  of  28  =  18  is  75%  of  — 

6.  66f  per  cent  of  21  =  18  is  66f  %  of  - 

7.  40  per  cent  of  60  =  20  is  40%  of  — 

8.  60  per  cent  of  60  =  24  is  60%  of  — 
d.  75  per  cent  of  32  =  27  is  75%  of  — 

10.  66f  per  cent  of  36  =  22  is  66|%  of  . 

(5)  {3) 

11.  8  is  per  cent  of  32.  8  is  %  of  24. 

12.  8  is  per  cent  of  40.  8  is  %  of  16. 

13.  8  is  per  cent  of  64.  8  is  %  of  56. 

14.  8  is  per  cent  of  48.  8  is  %  of  72. 

15.  8  is  • per  cent  of  80.  8  is  %  of  20. 

16.  9  is  per  cent  of  15.  18  is  %  of  27. 

17.  18  is  per  cent  of  24.  4-1-  is  %  of  13. 

18.  Eussel  earned  60^  ;  lie  spent  10%  of  his  money  for  a 

tablet  and  20%  of  it  for  a  book;  the  tablet  cost  cents 

and  the  book  cost  cents. 

19.  Ten  per  cent  of  the  sheep  in  a  certain  flock  were 

black ;  there  were  8  black  sheep ;  in  all  there  were  

sheep. 


68  COMPLETE    ARITHMETIC. 

PERCENTAGE. 

(1) 

1.  40  per  cent  of  55  =  60  per  cent  of  55  = 
(a)  Find  40%  of  575.  (b)  Find  60%  of  365. 
(c)  Find  75%  of  676.  (d)  Find  66f  %  of  591. 
(e)  Find  3%  of  $254.  (f)  Find  7%  of  $254. 

(g)  A  lawyer's  commission  for  collecting  money  was 
7%  ;  he  collected  $635.  How  much  of  the  money  should 
he  keep  and  how  much  should  he  "  pay  over "  to  the  man 
for  whom  he  collected  the  money  ? 

(^) 

2.  18  is  40%  of  .  18  is  60%  of  . 

(h)  346  is  40%  of  what  ?      (i)   345  is  60%  of  what  ? 

(j)  534  is  75%  of  what  ?      (k)  534  is  66|%  of  what? 

(1)  $36  is  3%  of  what?        (m)$84  is  7%  of  what? 

(n)  A  lawyer's  commission  for  collecting  money  was 
7%  ;  his  commission  amounted  to  $63.  How  much  did 
he  collect  and  how  much  should  he  "  pay  over  "  to  the  man 
for  whom  he  collected  the  money  ? 

(5) 

3.  18  is  per  cent  of  45.  33  is  %  of  55. 

4.  33  is  per  cent  of  44.  18  is  %  of  27. 

(o)  Eighteen  dollars  are  what  per  cent  of  $600  ?  This 
mesiiis,  find  how  many  hundredths  of  $600,  $18  ao^e. 

Operation.                                    Explanation. 
$600  -^  100  =r  $6.  One  per  cent  of  $600  is . 

$18  -^  $6  =  3  times.         ^^^  ^''^  ''^  "'^'^^  P^'*  ''^"^  "!*  ^^^^ 
^    /      o  ^/.^^         as  t^6  are  contained  times  in  $18. 
$18  are  3%  of  $600. 

6.  $84  are %  of  $1200.     $72  are  %  of  $800. 


PART   I.  59 


REVIEW. 


1.  The  prime  factors  of  63  are  ,  — -,  and  . 

(a)  What  are  the  prime  factors  of  215  ?     (b)  Of  470  ? 

2.  Three,  3,  3,.  and  2  are  the  prime  factors  of  . 

(c)  Of  what  number  are  3,  5,2,  and  23  the  prime  factors  ? 

3.  Add  I  and  |.     The  1.  c.  m.  of  8  and  3  is  . 

(d)  Add  I,  -A,  and  |.     The  1.  c.  m.  of  9,  12,  and  3  is 


(e)  Divide  |  by  -^.     The  1.  c.  m.  of  9  and  12  is  ■ 

4.  Multiply  $324  by  |.     This  means  . 

(f)  Multiply  $324  by  .25. 

5.  One  half  of  a  rod  is  and feet. 


(g)  Twenty-six  and  one  half  rods  are  how  many  feet  ? 

6.  Three  eighths  of  a  mile  are  rods. 

(h)  How  many  rods  in  7|  miles  ?      (i)  In  4|-  miles  ? 

7.  Two  rods  are  — —  yards.     3  rd.  are  yards. 

(j)  How  many  yards  in  42  rods  ?    (k)  In  64  rods  ? 

8.  When  wheat  is  70^  a  bushel,  240  lb.  cost  . 

(1)  When  wheat  is  90^  a  bushel,  how  much  will  one  ton 
of  wheat  cost  ? 

9.  A  pile  of  wood  32  feet  long,  4  feet  wide,  and  4  feet 
high  contains  cords. 

(m)  How  many  cu.  ft.  in  a  pile  of  wood  32  ft.  by  4  ft.  by 
2  f t.  ? 

10.  If  12  lb.  of  sugar  are  worth  45^,  at  the  same  rate 
8  lb.  are  worth  cents. 

(n)  If  12  boxes  of  soap  are  worth  $22.50,  how  much  are 
8  boxes  worth  at  the  same  rate  ? 


60  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    PROBLEMS. 

1.  The  perimeter  of  a  rectangular  surface  7  feet  by  9 
feet  is  feet. 

(a)  How  many  rods  of  fence  are  required  to  enclose  a 
rectangular  field  38  rods  by  54  rods  ? 

2.  If  the  sun  rises  at  7  o'clock  and  sets  at  4:30,  from 
sunrise  to  sunset  it  is  hours  and  minutes. 

(b)  If  the  sun  rises  at  7:14  and  sets  at  4:28,  how  long  is 
it  from  sunrise  to  sunset  ? 

3.  A  man  bought  16  dozen  eggs  at  10^  a  dozen;  he  lost 
25  per  cent  of  them  by  decay;  he  sold  the  remainder  at  12^ 
a  dozen ;  he  lost  cents. 

(c)  A  man  bought  360  dozen  eggs  at  9^  a  dozen ;  he  lost 
25%  of  them  by  decay;  he  sold  the  remainder  at  11^  a 
dozen.     How  much  money  did  he  lose  ? 

4.  From  midnight  Monday  to  midnight  Wednesday  it 
is  hours. 

(d)  How  many  hours  in  a  week  ?  (e)  In  the  month  of 
January  ? 

5.  At  45^  each,  3  copies  of  "Eobinson    Crusoe"  wiU 

cost  dollar  and  cents;  for  $1.80  I  can  buy  

copies. 

(f)  How  much  will  7  copies  of  "  Robinson  Crusoe  "  cost  ? 

(g)  How  many  copies  of  "  Robinson  Crusoe "  can  I  buy 
for  $10.35  ?       (h)  For  $4.95  ? 

(i)  Add  four  thousand  three  hundred  twenty-four  and 
twenty-five  thousandths,  and  seven  hundred  forty-six  and 
thirty-four  hundredths,  and  seventy-five  and  eight  tenths. 

(j)  From  forty-five  hundred  subtract  forty-five  hundredths. 


PART    I.  61 

SIMPLE    NUMBERS. 

1.  Ten  times  60  are  .  10  times  40  are  . 

2.  Ten  times  46  are  .  10  times  25  are  . 

3.  Ten  times  300  are  .         10  times  325  are  . 

4.  Ten  times  2  times  a  numher  are times  the  number. 

(a)  Multiply  347  by  20. 

Operation.  Explanation. 

347  Two  times  347  are  . 

20  Ten  times  694  are  . 


fiQAO  ^^^  times  2  times  (20  times)  347  are  . 

(b)  Multiply  564  by  30.         (c)  Multiply  468  by  40. 
(d)  Multiply  735  by  50.         (e)  Multiply  642  by  60. 

5.  One  hundred  times  2  times  a  number  are  times 

the  number. 

(f)  Multiply  86  by  200.         (g)  Multiply  94  by  300. 

(h)  Multiply  75  by  400.         (i)  Multiply  48  by  700. 

REVIEW. 

6.  The  prime  factors  of  48  are  ,  ,  ,  , 


and  . 

(j)  What  are  the  prime  factors  of  290  ?  (k)  Of  430  ? 

Multiply.  Divide.  Divide. 

(1)  347  by  70.    (m)  8540  lb.  by  70  lb.*    (n)  1480  lb.  by  20.t 
(o)  575  by  70.    (p)  9470  lb.  by  70  lb.      (q)  1350  lb.  by  20. 

*  This  means,  find  how  many  times  70  lb.  are  contained  in  8540  lb.  70  lb.  are 
contained  in  8540  lb.  times.    Story^In  85U0  lb.  of  ear  corn  there  are btishels. 

tThis  means,  find  1  twentieth  of  1480  lb.  1  twentieth  of  1480  lb.  is lb.   Story 

—Iwish  to  put  U80  lb.  of  wheat  into  20  sacks;  I  must  put lb.  in  each  sack. 


62  COMPLETE    ARITHMETIC. 

COMMON   FKACTIONS. 

1.  Add  3V  and  -^  *  (a)  S75^-{-256^\. 

2.  From  7|  subtract  2ft    (b)  3467^  -  12824. 

3.  Divide  I  by  3V.  (c)  3| -^  ^.  «     (9) 

4.  Divide  8  by  |.  (d)  28  -i- 1.  (4) 

5.  Divide  3f  by  If  (e)  47^  -^  1|.  (h) 

6.  Divide  21^  ft.  by  3.  (f)  87|  ft.  -^  3.  (21) 

7.  Divide  1|  ft.  by  3.  (g)  5|-  ft.  ^  3.  (15) 

8.  Divide  13|-  ft  by  3.  (h)  745f  ft.  ^  3.  (21) 

9.  Multiply  20^  by  2f.t      (i)   96  x  2f.  (I6) 

10.  Multiply  20^'  by  |.  (j)   124  x  J.  (8) 

11.  Multiply  $f  by  6.  (k)  64f  x  6.  (19) 

12.  Multiply  $1-  by  J.  (1)  $i  x  26f  (13) 

13.  Multiply  $1^  by  i.  (m) $27|-  x  |.  (13) 

14.  Multiply  $61^  by  2f  §  (n)  $86|-  x  2  J.  (22) 

15.  At  $4|^  per  ton,  2  tons  of  coal  cost  dollars;  ^  a 

ton  costs  ;  2i  tons  cost . 

16.  At  $8^  per  ton,  2|-  tons  of  hay  cost  . 

*  The  pupil  is  expected  to  find  by  trial  that  the  1.  e.  m.  of  10  and  12  is . 

fTake  1  from  the  7,  change  it  to  fifths  and  add  it  to  the  J. 
t  In  problems  9  to  13  inclusive,  think  of  the  number  to  be  multiplied  as  the  price 
per  yard;  this  will  suggest  the  following: 

Problem  9,  21  yd,  of  ribbon  at  20^  a  yard. 
Problem  10,  J  yd.  of  ribbon  at  20^  a  yard. 
Problem  11,  6  yd.  of  ribbon  at  $1  a  yard. 
Problem  12,  J  yd.  of  ribbon  at  $J  a  yard. 
Problem  13,  i  yd.  of  ribbon  at  8U  a  yard. 

g  This  means,  2  times  86^  plus  J  of  86i.    2  times  S6i  are dollars,    i  of 

86i  is dollars.    $13  +  $31  = dollars.    Stoi-y— At  $6i  a  ton,  2i  tons  of 

coal  will  cost .    i  See  note,  page  6. 


PART   I.  63 

DECIMAL    FRACTIONS. 

(a)  Multiply  $546  by  3.24.     This  means,  find  3  times 
$546,  plus  2  tenths  of  $546,  plus  4  hundredths  of  $546. 

Operation.  Explanation. 

$546  One  hundredth  of  |546  is  . 

3.24  ^  hundredths  of  $546  are  . 

One  tenth  of  $546  is  . 

^^^•^^  2  tenths  of  $546  are  . 

$109.2  3  times  $546  are  . 

$1638  $21.84  +  $109.2  +  $1638  = 
$1769.04 

NUMBER   STORY. 

K  one  acre  of  land  is  worth  $546, 

1  hundredth  of  an  acre  is  worth  . 

4  hundredths  of  an  acre  are  worth  . 

1  tenth  of  an  acre  is  worth  . 

2  tenths  of  an  acre  are  worth  . 

3  acres  are  worth  . 

3.24  acres  are  worth  . 

(b)  Multiply  $437  by  2.36  ;  (c)  $375  by  5.27. 
(d)  Multiply  $352  by  6.21 ;  (e)  $284  by  7.32. 

(f)  Multiply  four  hundred  seventy-three  dollars  by  four 
and  thirty-five  hundredths. 

(g)  When  oil  meal  is  $28  a  ton,  how  much  will  4680  lb. 
cost  ?     (4680  lb.  = tons.) 

(h)  (i)                 (j)                     (k)                    (1) 

Add.  Subtract.  Multiply.  Divide.  Divide. 

.886  8.744              8.2*  $.5)$31              5)$31. 

.075  .956               M_ 

*  One  hundredth  of  8.2  is  .082.    The  pupil  should  write  the  decimal  jwint  in  the 
first  partial  product  immediately  after  xvnting  the  tenths'  figure  oj  (he  partial  product. 


64  COMPLETE   ARITHMETIC. 

DENOMINATE    NUMBERS. 
]Sfn      ^^^  CITY    SCALE. 

Load  of E^ Dec.gj,         -1^9^ 

^^^^ Clyde  H.  Hall rp^        Fay  D.  Winslow. 

.     '  Gross  Weight, ^J^ Ih. 

Tare, ^1^ Ih. 

Net  Weight, ???? Ih. 

Clarence  Marshall,         Weiaher 

(a)  Find  the  value  of  the  load  of  hay,  ticket  No.  186,  at 
$8.25  a  ton. 

Find  the  value  of  the  following : 

Commodity.       Gross  Weight.  Tare.  Price  per  Ton. 

(b)  Hay.  5610  1870  $12.50 

(c)  Coal.  6380  2140  $6.50 

(d)  Bran.  3340  1560  $13.50 


1.  One  hundred  feet  are  rods  and  foot. 

2.  Two  hundred  feet  are  rods  and  feet. 

3.  Three  hundred  feet  are  rods  and  feet. 

4.  Four  hundred  feet  are  rods  and  feet. 

5.  Five  hundred  feet  are  rods  and  feet. 

(e)  Change  8  miles  to  rods,    (f)  Change  30  rods  to  feet, 
(g)  Change  36  rods  to  yards,  (h)  Change  2160  lb.  oats  to  bu. 
(i)  Change  5240  lb.  to  tons,  (j)  Change  274  ft.  to  inches. 


PART    I.  65 


MEASUREMENTS. 

1.  One  hundred  sixty  sqiiare  rods  are  one  acre.     A  piece 
of  land  1  rod  wide  and  160  rods  long  is  one  r-. 

2.  A  piece  of  land  i  of  a  mile  long  and  1  rod  wide  is 


3.  Land  2  rd.  wide  and  rd.  long  is  one  acre. 

4.  Land  4  rd.  wide  and  rd.  long  is  one  acre. 

5.  Land  8  rd.  wide  and  rd.  long  is  one  acra. 

6.  Land  10  rd.  wide  and  • rd.  long  is  one  acre. 

7.  Land  5  rd.  wide  and  rd.  long  is  one  acre. 

8.  Land  4  rods  by  4  rods  is  th  of  an  acre. 

9.  Land  4  rods  by  10  rods  is  - —  of  an  acre. 

10.  Land  8  rods  by  10  rods  is  of  an  acre. 

11.  Land  1  rod  wide  and  1  mile  long  is  — —  acres. 

12.  Land  2  rods  wide  and  1  mile  long  is  acres. 

13.  Land  66  ft.  wide  and  1  mile  long  is  acres. 

14.  Land  5  rods  by  16  rods  is  square  rods. 

(a)  How  many  square  rods  in  a  rectangular  piece  of  land 
28  rd.  by  36  rd.? 

REVIEW. 

(b)  Find  the  area  of  a  rectangular  surface  18  ft.  by  26  ft.* 

(c)  Find  the  volume  of  a  rectangular  solid  15  ft.  by  8  ft. 
by  23  ft.f 

15.  How  many  cords  in  a  pile  of  wood  4  ft.  wide,  4  ft. 
high,  and  36  ft.  long? 

(d)  How  many  cords  in  a  pile  of  wood  4  ft.  high,  4  ft. 
wide,  and  276  ft.  long? 

(e)  How  many  cords  in  a  pile  of  wood  4  ft.  wide,  4  ft. 
high,  and  as  long  as  your  schoolroom  ? 

*  See  foot-note  (*),  page  55.       f  See  foot-note  (f),  page  55. 


COMPLETE    ARITHMETIC. 


RATIO   AND   PROPORTION. 


1.  One  ninth  of  36  is  .  36  is  ^  of  . 

2.  One  eighth  of  36  is  .  36  is  i  of  . 

3.  One  seventh  of  36  is  .  36  is  -i-  of  . 

(a)  Find  1  eighth  of  353.6.  (b)  353.6  is  i  of  what? 

(c)  Find.  1  fifth  of  1685.  (d)  1685  is  i  of  what? 

4.  Five  eighths  of  80  are  .         80  is  f  of  . 

5.  Three  fifths  of  60  are  .  60  is  f  of  . 


(e)  Find  |-  of  680.         (f)  680  is  |  of  what  number  ? 
(g)  Find  I  of  435.         (h)  435  is  |  of  what  number  ? 

6.  22  is of  33.  33  is of  22. 

7.  33  is of  55.  55  is of  33. 

8.  55  is of  99.  99  is of  55. 

9.  44  is  — _  of  77.  77  is of  44. 

10.  Three  eighths  of  32  are  2  thirds  of  .    ' 

11.  Five  eighths  of  32  are  2  thirds  of  . 


(i)    Three  eighths  of  528  are  2  thirds  of  what  number? 
(j)    Five  eighths  of  528  are  2  thirds  of  what  number? 

12.  Twenty  is  of   16,  or  and  — 


times  16.     A  man  can  earn  and times  as 

much  in  20  days  as  he  can  earn  in  16  days.    If  he  can  earn 

$44  in  16  days,  in  20  days  he  can  earn  dollars. 

(k)  If  a  man  can  earn  $740  in  16  weeks,  how  much  can 
he  earn  in  20  weeks  ? 

13.  If  20  bushels  of  apples  are  worth  $12,  15  bushels  are 
worth  dollars. 

(1)  If  20  gal.  of  milk  are  worth  $3.24,  how  much  are  15 
gal.  worth  ? 


PART    I.  67 

PEBCENTAGE. 

80  per  cent  =  .80  =  f 
831^  per  cent  =  .83^-:  |. 

W  (^) 

1.  80  per  cent  of  60  =  60  is  80%  of  . 

2.  83i  per  cent  of  60  -  30  is  83^%  of  , 

3.  80  per  cent  of  40  =  40  is  80%  of  . 

4.  83|-  per  cent  of  36  ■=  35  is  83^%  of  . 

5.  75  per  cent  of  60  =  60  is  75%  of  . 

6.  66f  per  cent  of  18  =  32  is  66f  %  of  . 

7.  40  per  cent  of  35  =  18  is  40%  of  . 

8.  60  per  cent  of  35  =  27  is  60%  of  . 

(3)  (3) 

9.  9  is  per  cent  of  54.  9  is  %  of  45. 

10.  9  is  per  cent  of  63.  9  is  %  of  36. 

11.  9  is  per  cent  of  27.  9  is  %  of  72.    ' 

12.  9  is  per  cent  of  18.  9  is  %  of  90. 

13.  9  is  per  cent  of  81.  14  is  %  of  35. 

14.  21  is  per  cent  of  35.  14  is  %  of  21. 

15.  21  is  *  per  cent  of  28.  35  is  %  of  42. 

16.  28  is  per  cent  of  35.  12  ^  is  %  of  25. 

17.  Mr.  Dow  had  80  bushels  of  apples;  he  lost  25%  of 

tham  by  decay ;  he  lost  bushels  and  had  bushels 

left. 

18.  Fourth-of-July  night  Willie  had  10^;  this  was  12|^ 
per  cent  of  what  his  father  gave  him  to  spend ;  his  father 
gave  him  cents. 

19.  Sarah  had  45  chicks;  a  hawk  killed  18  of  them; 
the  hawk  killed  per  cent  of  her  chickens. 


68  COMPLETE    ARITHMETIC. 

PERCENTAGE. 

1.  80  per  cent  of  55  =  83^  per  cent  of  48  = 
(a)  Find  80%  of  435.  (b)  Find  83-i-%  of  492. 
(c)  Find  40%  of  435.  (d)  Find  75%  of  492. 
(e)  Find  7%  of  $435.  (f)  Find  9%  of  $492. 

(g)  A  lawyer's  commission  for  collecting  money  was  9  %;* 
he  collected  $834.  How  much  of  this  money  should  he 
keep  ?  and  how  much  should  he  "  pay  over  "  to  the  man  for 
whom  he  collected  the  money  ? 

(^) 

2.  20  is  80%  of  .  20  is  83^  %  of . 

(h)  280  is  80%  of  what  ?        (i)   280  is  83^%  of  what  ? 

(j)  276  is  60%  of  what?        (k)  276  is  75%  of  what? 

(1)   224  is  7%  of  what  ?  (m)  531  is  9%  of  what  ? 

(n)  A  lawyer's  commission  for  collecting  money  was  9  %  ;* 
his  commission  amounted  to  $54.  How  much  did  he  col- 
lect ?  and  how  much  should  he  "  pay  over  "  to  the  man  for 
whom  he  collected  the  money  ? 

(5) 

3.  Fifty-six  dollars  are  what  per  cent  of  $800  ?  This 
means,  $56  are  how  many  hundredths  o/  $800  ? 

4.  $36  is  %  of  $400.  $55  is  %  of  $500. 

5.  $2.40  is  %  of  $80.  $3.50  is %  of  $50. 

6.  A  lawyer  collected  $900  ;  he  retained  as  his  commis- 
sion $63  of  this  sum,  and  paid  the  remainder,  $837,  to  the 
man  for  whom  he  collected  the  money ;  the  lawyer's  com- 
mission for  collecting  was  %. 

*  %  of  what  ?    95t  of  the  amount  collected. 


PART    I.  69 

REVIEW. 

1.  Ten  times  6  times  a  number  are  times  the  num- 
ber.    10  times  6  times  8  are  .     8  x  60  = 

2.  One  hundred  times  5  times  a  number  are  times 

the  number.     9  multipHed  by  500  = 

(a)  Multiply  78  by  80.     (b)  Multiply  96  by  700. 

3.  Divide  12^  hj  2i.     (Change  to  halves.)     Story. 

(c)  Divide  345|  by  2f.     (Change  to  .)     Stori/. 

(d)  Divide  345.6  by  2.4.     This  means,  .     Story. 

(e)  Multiply  $338  by  2.4.     (f)  Multiply  $338  by  2.43. 
(g)  When  land  is  $375  an  acre,  how  much  will  3.35  acres 

cost  ? 

Find  the  value  of  the  following : 

Commodity.  Gross  Weight.  Tare.  Price  per  Ton. 

(h)  Straw.  4360  1b.  1780  1b.  $4.25 

(i)  Coal.  6240  lb.  1830  lb.  $6.75 

4.  One  acre  is  square  rods.         ^  acre  = 

5.  One  mile  is  rods.  ^  mile  = 

6.  A  piece  of  land  ^  of  a  mile  long  and  2  rods  wide  is 
acres. 

7.  A  piece  of  land  160  rods  long  and  as  wide  as  the 
schoolroom  is  about  acres. 

(j)  How  many  square  rods  in  a  rectangular  piece  of  land 
47  rods  by  6  rods  ?  Is  this  more  or  less  than  1  acre  ?  Is  it 
more  or  less  than  2  acres  ? 

8.  One  cord  is  cubic  feet.     ^  cord  = 

(k)  How  many  cubic  feet  of  wood  in  a  pile  7  feet  by  5 
feet  by  6  feet  ?  Is  this  more  or  less  than  1  cord  ?  Is  it 
more  or  less  than  2  cords  ? 


70  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    PROBLEMS. 

1.  If  a  man  can  save  $5  a  month,  in  3  years  he  can 
save  dollars. 

(a)  If  a  man  can  save  $2  7  in  a  month,  how  much  can  he 
save  in  9  years  ? 

2.  Byron  bought  2  doz.  oranges  for  40^;  he  sold  them 
at  3^  each ;  he  gained  cents. 

(b)  A  merchant  bought  35  barrels  of  apples  for  $85 ;  he 
sold  them  at  $2.75  a  barrel.     How  much  did  he  gain  ? 

3.  Henry  had  25   chickens;    a  hawk  caught  20%    of 

them;  he  sold  the  remainder  at  22^  each;  he  received  

dollars  and  cents. 

(c)  Henry's  father  had  75  bushels  of  apples ;  he  lost  20% 
of  them  by  decay;  he  sold  the  remainder  at  85^  a  bushel. 
How  much  did  he  receive  for  them  ? 

4.  If  coffee  costs  $f  a  pound,  for  1  dollar  I  can  buy 
pounds.     1^1  = 


(d)  If  tea  costs  $|  a  pound,  how  many  pounds  can  I  buy 
for  $57?      (Change  $57  to  fifth-dollars.) 

6.  The  sum  of  two  numbers  is  34 ;  one  of  the  numbers 
is  12  ;  the  other  number  is  . 

(e)  The  sum  of  two  numbers  is  346.2 ;  one  of  the  num- 
bers is  75.36.     What  is  the  other  number? 

6.  In  a  piece  of  slate  1  foot  square  and  1  inch  thick 
there  are  cubic  inches. 

(f)  How  many  cubic  inches  in  a  piece  of  slate  2  feet 
square  and  2  inches  thick  ? 

(g)  If  |-  of  a  ton  of  coal  is  worth  $4.20,  how  much  is  1 
ton  worth  ? 


PAKT   I.  71 


SIMPLE    NUMBERS. 

1.  One  tenth  of  6  is  .  1  tenth  of  40  is  - 

2.  One  tenth  of  46  is  .         1  tenth  of  25  is  - 

3.  One  tenth  of  300  is  .      1  tenth  of  325  is 


4.  One  half  of  1  tenth  of  a  number  is  one  th  of  the 

number.     One  third  of  1  tenth  of  a  number  = 

(a)  Divide  472  by  20. 

Operation.                                        Explanation. 
20)47  "^2  C)ne  tenth  of  472  is  . 

— TT-T-  One  half  of  1  tenth  (^)  of  472  is  . 

Zo.o 

(b)  Divide  741  by  30.      .   (c)  Divide  548  by  40. 
(d)  Divide  735  by  50.  (e)  Divide  960  by  60. 

5.  One  third  of  1  hundredth  of  a  number  is  one  th  of 

the  number.     One  fourth  of  1  hundredth  of  a  number  = 

(f)  Divide  972  by  300.       (g)  Divide  972  by  400. 
(h)  Divide  895  by  500.       (i)  Divide  976  by.  800. 

REVIEW. 

6.  Common  multiples  of  15  and  6  are  ,  ,  etc. 

The  least  common  multiple  of  15  and  6  is  -^ — . 

7.  The  prime  factors  of  63  are  ,  ,  and . 

(j)  What  are  the  prime  factors  of  124  ?     (k)  Of  178  ? 

8.  3,  3,  and  5  are  the  prime  factors  of  .     Forty-five 

is  exactly  divisible  by  3  ;  by  5  ;  by  ;  by  . 

(1)  ■  (m) 

128  cu.  feet.)1536  cu.  ft.  12)1584  cu.  ft. 

(n)  (o) 

160  sq.  rd.)2400  sq.  rd.  15)2445  sq.  rd. 


72  COMPLETE    ARITHMETIC. 


COMMON    FRACTIONS. 

1.  Add  y3_  and  J.     The  1.  c.  m.  of  15  and  6  is  - 

(a)  Find  the  sum  of  456^-^,  341f,  245|,  and  564. 

2.  From  9^?^  subtract  4|.     1^-^  =30.     i=  to- 

(b)  Find  the  difference  of  4275^2-^.  and  1328|. 

3.  Multiply  I  by  9.     This  means  3*  . 


(c)  Find  the  product  of  453-|-  multiplied  by  9. 

4.  Multiply  2f  by  -i.     This  means  13* 

(d)  Find  the  product  of  45 8  J  multiplied  by  i 

5.  Multiply  17  by  |.     This  means  8*  - 


(e)  Find  the  product  of  741  multiplied  by  |. 

6.  Multiply  17  by  2f.     This  means  16*  , 

(f)  Find  the  product  of  741  multiplied  by  2|. 

7.  Multiply  16i  by  2^.     This  means  - —  22*  — 

(g)  Find  the  product  of  732^-  multiplied  by  2i. 

8.  Divide  8  by  f .     (4)*     Change  8  to    ths. 

(h)  Find  the  quotient  of  97  divided  by  f.     Stori/. 

9.  Divide  ^  by  -J.     (9)*     Change  to  ^ths. 

(i)  Find  the  quotient  of  3^1  divided  by  J.     Story. 

10.  Divide  7|  by  2^.     (14)*     Change  to  ths. 

(j)    Find  the  quotient  of  55^  divided  by  2f.     Sto7y. 

11.  Divide  Slf  (|)  by  4.-    This  means  15*  — 


(k)  Find  the  quotient  of  7f  divided  by  8.     Story. 

a)  (^)  (^)  (^)  (P) 

Add.       Subtract.    Multiply.  Divide.  Divide. 

375ff    4351     346|    3|  ft.)232  ft.    3)365 1  ft. 

2465     182|^    12 


*  These  figures  refer  to  notes  on  pages  6  and  7.    See  also  foot-notes,  page  i 


PART    I.  73 

DECIMAL    FRACTIONS. 

1.  One  tenth  of  S6  is  .         .1  of  $6.25  is  . 


2.  One  hundredth  of  $6  is .       .01  of  $6.25  is  $.0625. 

3.  Eead  each  of  the  following  in  two  ways:  $.2436,* 
$.0532,  $.6403,  $.0042,  $.0002,  $.6042,  $.8002. 

(a)  Multiply  $6.25  by  4.23.     This  means,  find  4  times 
$6.25  +  2  tenths  of  $6.25  +  3  hundredths  of  $6.25. 
Operation.  Explanation. 

$6.25  One  hundredth  of  $6.25  is  . 

4.23  3  hundredths  of  $6.25  are  . 

<m  1  oycj.  One  tenth  of  $6.25  is  . 

$1250  "  tenths  of  $6.25  are  . 

4  times  $6.25  are  . 


$25.00 
$26.4375 


$.1875 +$1.250 +  $25  = 


NUMBER    STORY. 


If  one  ton  of  coal  is  worth  $6.25, 
1  hundredth  of  a  ton  is  worth  . 

3  hundredths  of  a  ton  are  worth  . 

1  tenth  of  a  ton  is  worth  . 

2  tenths  of  a  ton  are  worth  . 

4  tons  are  worth  . 

4.23  tons  are  worth  . 

(b)  Multiply  $7.35  by  3.46 ;     (c)  $4.45  by  5.24. 

*  (1)  24^,  3  m.,  and  6  tenths  of  a  mill.    (2)  2436  ten-thousandths  of  a  dollar. 

f  To  THE  Teacher. — Read  the  foot-note  on  page  133 ;  also,  the 
first  part  of  page  143.  If  the  pupil  finds  difficulty  in  "  pointing  off," 
teach  him  to  use  a  separatrix  in  the  multiplicand  as  suggested  on 
page  133.     While  multiplying  6.25  by  3  hundredths,  it  may  appear 

"^06.25 
on  the  slate  thus :        ^9;^     Do  not  at  this  stage  of  the  work  allow 


the  pupil  to  "  point  oft" "  by  counting  the  decimal  places  in  the  mul- 
tiplicand and  multiplier.  Rather,  lead  him  to  think  the  meaning 
of  the  problem.     See  foot-notes,  page  53. 


74 


COMPLETE    ARITHMETIC. 


DENOMINATE   NUMBERS. 


1.  5280  feet  are  1  mile.     - 
(a)  Change  32*0  rd.  to  feet. 

2.  Six  hundred  feet  are  — 
3. 
4. 

5. 


Seven  hundred  feet  are  - 
Eight  hundred  feet  are  - 

Two  miles  are  rods. 


—  rods  are  1  mile. 

(b)  Change  160  rd.  to  feet. 

rods  and  feet. 

—  rods  and  feet. 

—  rods  and  feet. 

1^  miles  are  rods. 


(c)  Change  6  mi.  to  rods, 
(e)  Change  2  mi.  to  feet. 
(g)  Change  2^  mi.  to  feet. 

Find  the  value  of  the  following  : 

Commodity.  Gross  Weight, 

(i)  Corn  meal.  4380  lb. 

(j)   Oat  straw.  3460  lb. 

(k)  Old  iron.  3240  lb. 

(1)  Paper-rags.  3120  lb. 

(m)  Ear  corn.  3690  lb. 

(n)  Oats.  2410  lb. 

6.  120  inches  are  feet. 


(o)  Change  428  in.  to  ft. 

7.  72  feet  are  yards. 

(q)  Change  853  ft.  to  yd. 

8.  33  feet  are  rods. 

(s)  Change  627  ft.  to  rd.f 

9.  11  yards  are  rods. 


(d)  Change  30  rd.  to  feet, 
(f)  Change  75  rd.  to  yards, 
(h)  Change  2|-  mi.  to  rods. 

Tare.  Price. 

1540  lb.  $15.25  per  ton. 

1620  lb.  $4.50  per  ton. 

1380  1b.  ^^  per  lb.* 

1260  1b.  1^  per  lb. 

1230  1b.  35^perbu. 

1250  1b.  28^perbu. 

126  inches  are  feet. 

(p)  Change  580  in.  to  ft. 

73  feet  are  yards. 

(r)  Change  725  ft.  to  yd. 

66  feet  are  rods. 

(t)  Change  594  ft.  to  rods. 

22  yards  are  rods. 


(u)  Change  759  yd.  to  rd.  (v)  Change  638  yd.  to  rd. 

*1860  lbs  at  1^  a  pound  would  be  worth  818.60 ;  at  J  a  cent  a  jwund  the  value  is 
I  of  S18.60,  or . 

tTo  divide  627  ft.  by  16 J  ft.,  change  both  numbers  to  halves.  See  page  7,  npte 
(20). 


PART    I.  75 

MEASUREMENTS. 

1.  A  piece  of  land  10  rods  by  16  rods  contains  

square  rods.     It  is  acre. 

2.  A  piece  of  land  20  rods  by  16  rods  contains  

square  rods.     It  is  •  acres. 

(a)  Change  640  sq.  rd.  to  acres,  (b)  Change  1280  sq. 
rd.  to  acres,  (c)  How  many  acres  in  a  piece  of  land  40 
rods  by  64  rods  ? 

3.  Eighty  square  rods  are of  an  acre. 

4.  Forty  square  rods  are of  an  acre. 

5.  One  hundred  twenty  square  rods  are of  an 

acre. 

6.  A  piece  of  land  20  rd.  by  26  rd.  contains  square 

rods. 

(d)  How  many  acres  in  a  piece  of  land  20  rd.  by  26  rd.  ? 

Find  the  number  of  acres  in  each  of  the  following : 

(e)  20  rods  by  28  rods.  (f)  20  rods  by  30  rods, 
(g)  12  rods  by  60  rods.  (h)  24  rods  by  50  rods, 
(i)  36  rods  by  50  rods.         (j)   48  rods  by  60  rods. 

7.  A  pile  of  wood  8  ft.  by  4  ft.  by  4  ft.  contains  

cubic  feet.     It  is  cord. 

8.  Sixty-four  cubic  feet  are of  a  cord. 

9.  Thirty-two  cubic  feet  are of  a  cord. 

(k)  How  many  cubic  feet  in  a  pile  of  wood  12  ft.  by  8  ft 
by  4  ft.  ? 

(1)  How  many  cords  in  a  pile  1 2  ft.  by  8  ft.  by  4  ft.  ? 

Find  the  number  of  cords  in  each  of  the  following : 

(m)  6  feet  by  4  feet  by  8  feet,     (n)  6  ft.  by  8  ft.  by  8  ft. 

(o)   6  feet  by  12  feet  by  4  feet,    (p)  6  ft.  by  12  ft.  by  8  ft 


76  COMPLETE   ARITHMETIC. 


RATIO   AND    PROPORTION, 

1.  One  fourth  of  3.6  is  *  3.6  is  ^  of  . 

2.  One  fifth  of  3.5  is  .  3.5  is  i  of  . 

(a)  Find  1  fourth  of  78.4.         (b)  78.4  is  ^  of  what  ? 
(c)  Find  1  fifth  of  97.5.  (d)  97.5  is  ^  of  what? 

3.  Three  fourths  of  3.6  are  .f  3.6  is  I-  of  — 


4.  Two  thirds  of  2.4  are  .  2.4  is  |  of  . 

(e)  Find  f  of  28.8.         (f)  28.8  is  |  of  what  number? 
(g)  Find  I  of  37.2.         (h)  37.2  is  |  of  what  number  ? 

5.  1.4  is  of  2.1.$         2.1  is of  1.4. 

6.  2.1  is of  2.8.  2.8  is of  2.1. 

7.  Two  thirds  of  2.1  are  1  half  of  . 

8.  One  half  of  2.4  is  2  thirds  of  . 

(i)    Two  thirds  of  53.7  are  1  half  of  what  number? 
(j)    One  half  of  65.6  is  2  thirds  of  what  number? 

9.  If  a  man  can  earn  $12  in  2.1  days,  in  2.8  days  he 
can  earn  dollars. 

(k)  If  a  man  can  earn  $75.45  in  2.1  months,  how  much 
can  he  earn  in  2.8  months  ? 

10.  If  2.8  tons  of  coal  are  worth  $12,  2.1  tons  are  worth 
dollars. 


(1)  If  2.8  tons  of  coal  are  worth  $9.60,  how  much  are 
2.1  tons  worth  ? 

11.  If  a  pile  of  wood  4  ft.  wide,  4  ft.  high,  and  8  ft.  long 
is  worth  $5,  a  pile  4  ft.  wide,  4  ft.  high,  and  32  ft.  long  is 
worth  dollars. 

*  One  fourth  of  36  tenths  is tenths. 

t  Three  fourths  of  36  tenths  are tenths,  or  2  and  tenths. 

i  Fourteen  tenths  are of  21  tenths. 


PART    I. 


77 


w 

1. 

121%  of  48  = 

2. 

25   %  of  48  = 

3. 

37i%  of  48  -^ 

4. 

50   %  of  48  = 

5. 

62^%  of  40  = 

6. 

871-%  of  56  = 

(5) 

7. 

3  is          %  of  24. 

8. 

21  is          %  of  24. 

9. 

4  is          %  of  32. 

10. 

12  is          %  of  32. 

11. 

5  is  %  of  40. 

PERCENTAGE. 

37^  per  cent  =  .37^  =  |. 
621  per  cent  =  .62-i-  =  |. 
87|-  per  cent  =  .87^  =  |. 

{2) 
48  is  121  %  of 
48  is  25  %  of 
48  is  37|-%  of 
48  is  50  %  of 
40  is  62^%  of 
56  is  87^%  of 


(5) 

— %  of  24. 
— %  of  24. 
— %  of  32. 
— %  of  32. 
— %  of  40. 


9  IS 
15  is 
20  is 
28  is 
15  is 


12.  Mr.  A  owed  Mr.  B  $72.  Mr.  A  had  suffered  many 
losses  and  it  was  hard  for  him  to  pay  this  amount.  Mr.  B 
kindly  discounted  the  bill  12|^%  and  Mr.  A  paid  it.  He 
paid  dollars. 

13.  Harry  paid  87^%  of  his  money  for  books  and  had  S2 

left ;  before  he  bought  the  books  he  had  dollars ;  the 

books  cost  dollars. 

14.  Mr.  Hill  set  40  trees  in  his  new  orchard ;  8  of  them 

died  the  first  year ;  per  cent  of  the  trees  were  dead ; 

per  cent  were  alive. 

15.  Twenty  per  cent  of  50  is  per  cent  of  30. 

16.  Forty  per  cent  of  50  is  per  cent  of  80. 


78  .  COMPLETE   ARITHMETIC. 

PERCENTAGE. 

{!) 

1.  37^  per  cent  of  64  =         62i  per  cent  of  64  = 
(a)  Find  37^%  of  192.         (b)  Find  62^%  of  192. 
(c)  Find  87^%  of  192.         (d)  Find  75%  of  192. 
(e)  Find  5%  of  $192.  (f)  Find  7%  of  $192. 

(g)  A  lawyer's  commission  for  collecting  money  was  8%;* 
he  collected  $875.  How  much  of  this  money  should  he 
keep  ?  and  how  much  should  he  "  pay  over  "  to  the  man  for 
whom  he  collected  the  money  ? 

(^) 

2.  15  is  37^  per  cent  of  .  35  is  62-i-%  of  . 

(h)  165  is  37J%  of  what?       (i)    165  is  62|-%  of  what? 
(j)  161  is  87|-%  of  what?       (k)  168  is  75%  of  what  ? 
(1)  135  is  3%  of  what?  (m)  168  is  7%  of  what? 

(n)  A  lawyer's  commission  for  collecting  money  was  8%;* 

his  commission  amounted  to  $34.80.  How  much  did  he 
collect  ?  and  how  much  did  he  "  pay-  over "  to  the  man  for 
whom  he  collected  the  money  ?  f 

(3) 

3.  18  is per  cent  of  48.      30  is per  cent  of  48. 

4.  Forty-five  dollars  is  what  per  cent  of  $500  ?  This 
means,  $45  are  hovj  many  hundredths  o/  $500  ?  J 

5.  A  lawyer  collected  $500 ;  he  retained  as  his  commis- 
sion $55  of  this  sum,  and  paid  the  remainder,  $445,  to  the 
man  for  whom  he  collected  the  money ;  the  lawyer's  com- 
mission for  collecting  was  %. 

*  8^  of  what  ?    %i  of  the  amount  collected. 

t  $34.80  is  8  hundredths  of  what  ? 

X  One  hundredth  of  $500  is  $5.    See  page  58,  problem  (o). 


PART    I.  79 


REVIEW. 


1.  One  fourth  of  1  tenth  of  a  number  is  one  th  of 

the  number.     1  fourth  of  1  tenth  of  120  is  . 

(a)  Divide  836  by  40.         (b)  Divide  75.6  by  40. 

2.  One  fifth  of  1  hundredth  of  a  number  is  one  th 

of  the  number.     1  fifth  of  1  hundredth  of  3500  is  . 

(c)  Divide  950  by  500.       (d)  Divide  246.5  by  500. 
(e)  Divide  3660  by  600.     (f)  Divide  5740  by  700. 

3.  Multiply  19  by  3|.        (This  means  .)* 

(g)  Multiply  276  by  5|.      (h)  Multiply  276  by  5.75. 
Compare  the  answers  to  problems  (g)  and  (h). 

4.  The  sum  of  86  hundredths  and  5  tenths  is  . 

(i)  Add  735  ten-thousandths  and  642  thousandths. 

5.  One  mile  is  feet.     1  mile  is  rods. 

(j)  Change  5i  mi.  to  feet,     (k)  Change  8|-  mi.  to  rods. 

6.  A  piece  of  land  4  rods  by  80  rods  is  acres. 

(1)  How  many  acres  in  a  piece  of  land  18  rd.  by  60  rd. 

7.  A  pile  of  wood  4  ft.  wide,  4  ft.  high,  and  40  ft.  long 
contains  cords. 

(m)  How  many  cords  in  a  pile  of  wood  4  ft.  wide,  4  ft. 
high,  and  432  ft.  long  ? 

Perform  the  operations  indicated  and  tell  number  stories : 
(n)  48  cu.  ft.  X  6  X  8  =      (o)  2304  cu.  ft.  ^  128  cu.  ft.=t 
(p)  96  sq.  rods  x  25  =       (q)  2400  sq.  rd.  -^160  sq.  rd.=$ 
(r)    960  rods  x  4  =  (s)  3840  rods  ^  320  rods  = 

(t)    2350  lb.  X  7  =  (u)  16450  lb.  -^  2000  lb.  = 

*  See  page  7,  note  16. 

t  Number  story  for  (n)  and  (o).  In  a  pile  of  wood  48  ft.  long,  6  ft.  wide,  and  8  ft. 
high  there  are cords. 

I  Number  story  for  (p)  and  (q).  A  rectangular  piece  of  land  96  rods  by  25  rods 
contains acre& 


80  COMPLETE   ARITHMETIC. 

MISCELLANEOUS   PKOBLEMS. 

1.  If  I  put  5.  oz.  of  candy  in  each  bag,  3  lb.  of  candy  will 
fill  bags,  with  ounces  over. 

(a)  If  I  put  2  bushels  of  oats  in  each  bag,  how  many  bags 
will  one  ton  of  oats  fill  ? 

2.  Henry  was  born  on  January  5, 1885;  on  Jan.  5, 1896, 
he  \vas  years  old. 

(b)  Queen  Victoria  was  born  on  May  24, 1819.  How  old 
was  she  on  the  24th  day  of  May,  1897  ? 

3.  A  dealer  bought  5  gal.  of  milk  at  14^  a  gal. ;  he  sold 
it  at  5^  a  quart,  but  lost  2  quarts  in  over-measurement  and 
waste ;  he  gained  cents. 

(c)  A  grocer  bought  3  bbl.  sugar,  each  containing  340  lb. ; 
he  paid  4^^*  a  lb.  If  he  sells  it  at  the  rate  of  20  lb.  for  a 
dollar,  but  loses  40  lb.  by  over-weights  and  waste,  how  much 
does  he  gain  ? 

4.  Edward  bought  2  lb.  candy  at  18^  a  pound,  2  lb. 
walnuts  at  20^  a  pound,  and  1  qt.  of  pop-corn  for  10^;  for 
all  he  paid  cents. 

Find  the  amount  of  each  of  the  following  bills: 

(d)  (e) 

3  lb.  Steak  @  15^  5  bu.  Apples  @  35^ 

^  bu.  Potatoes      @  40^  |-  bu.  Beans    @  $1.50 

18  lb.  Sugar  @  5^  7  gals.  Milk   @  15c. 

5  cans  Corn  @  8^  2  cd.  Wood    @  $4.50 

7  heads  Cabbage  @  8(f  1|  tons  Hay  @  $8.50 

5.  If  the  divisor  is  6  and  the  quotient  is  4 J,  the  dividend 
is  . 


(f)  If  the  divisor  is  128  and  the  quotient  16 J,  what  is 
the  dividend  ? 


PART   I.  81 

SIMPLE    NUMBERS. 

1.  Alfred  rode  his  bicycle  for  3  consecutive  hours ;  the 
first  hour  he  rode  1 0  miles  ;  the  second  hour,  8  miles  ;  the 

third  hour,  6  miles ;    altogether  he  rode   miles ;   his 

average  speed  was  (24  -r-  3)  miles  an  hour. 

2.  A  farmer  bought  20  sheep;  for  10  of  them  he  paid 

$35,  and  for  the  other  10,  $25  ;  for  all  he  paid dollars; 

the  average  cost  per  head  was  dollars. 

(a)  If  8  cows  cost  $204,  what  is  the  average  cost  per 
head? 

(b)  Five  boys  were  examined  in  spelling;  the  mark? 
received  were  90,  95,  85,  80,  and  75.  What  was  the  avei  - 
age  standing  of  the  boys  ? 

(c)  The  temperature  at  noon  of  each  day  for  1  week  was 
as  follows  :  Sunday,  94 ;  Monday,  91 ;  Tuesday,  96  ;  Wednes- 
day, 95;  Thursday,  90;  Friday,  89;  Saturday,  89.  What 
was  the  average  temperature  at  noon  for  the  week  ? 

(d)  Six  beef  cattle  weighed  as  follows  :  1560  lb.,  1430  lb., 
1640  lb.,  1350  lb.,  1420  lb.,  and  1660  lb.  What  was  the 
average  weight  per  head  ? 

REVIEW. 

3.  The  prime  factors  of  70  are  ■,  ,  and  . 

(e)  What  are  the  prime  factors  of  195  ?         (f)  Of  185  ? 

4.  2,  3,  and  7  are  the  prime  factors  of  .     Forty-two 

is  exactly  divisible  by  2 ;  by  3 ;  by  ;  by ;  by  , 

and  by  . 

(g)  Multiply  864  by  50  (h)  Divide  7650  by  50. 

(i)  Multiply  875  by  700.         (j)  Divide  8750  by  700. 
(k)  Multiply  736  by  400.         (1)  Divide  9384  by  400. 


82  COMPLETE    ARITHMETIC. 

COMMON    FRACTIONS. 

1.  Add  -f^  and  J.     The  1.  c.  m.  of  16  and  6  is  — 

(a)  Find  the  sum  of  376y\,  254|-,  321|,  and  342. 

2.  From  1-^^  subtract  3|.       l^-g  r.  ^^.       |  =  ^^. 

(b)  Find  the  difference  of  6274^ig  and  842|. 

3.  Multiply  -^^  by  8.     This  means . 

(c)  Find  the  product  of  357y^g  multiplied  by  8. 

4.  Multiply  If  by  ^.     This  means  . 

(d)  Find  the  product  of  573f  multiplied  by  \. 

5.  Multiply  19  by  f .     This  means  . 


(e)  Find  the  product  of  643  multiplied  by  f . 
6.  Multiply  19  by  2f.     This  means  


(f)  Find  the  product  of  643  multiplied  by  4|. 
7.  Multiply  15|-  by  2|.     This  means  — 


(g)  Find  the  product  of  345 -|-  multiplied  by  4i. 

8.  Divide  8  by  |.     (Change  8  to  ths.) 

(h)  Find  the  quotient  of  86  divided  by  f.     Story. 

9.  Divide  y'g  by  \ .     (Change  to  —ths.) 

(i)  Find  the  quotient  of  '^^^  divided  by  -\.     Story. 

10.  Divide  8|  by  2|.     (Change  to  ths.) 

(j)   Find  the  quotient  of  57|  divided  by  2|.     Story. 

11.  Divide  $1|-  (|)  by  5.     This  means . 

(k)  Find  the  quotient  of  8|  divided  by  9. 

(1)   Find  the  quotient  of  45 3|  divided  by  4. 

(m)  (n)                   (0)                  (p)  (q) 

Add.  Subtract.  Multiply.          Divide.  Divide. 

436|^  573^               245  3|  in.)246  in.  3)246 1  in. 

148|  245|^               24f 


PART    I.  83 

DECIMAL    FRACTIONS. 

Division.     Case  I. 

1.  Divide  .45  by  .09.     This  means  4* . 

(a)  Find  the  quotient  of  46.75  divided  by  .25.f     Stor/j. 

2.  Divide  9  by  .6.     This  means  9* . 

(b)  Find  the  quotient  of  874  divided  by  .5.|     Storij. 

3.  Divide  6  by  .05.     This  means  14* . 

(c)  Find  the  quotient  of  96  divided  by  .25.§     Stori/. 

4.  Divide  4.5  by  .09.     This  means . 

(d)  Find  the  quotient  of  46.5  divided  by  .25. |j      Stori/. 

(e)  Find  the  quotient  of  57.5  divided  by  2.5.^     Story. 

(f)  Find  the  quotient  of  68.5  divided  by  .25.**     Story. 

(g)  Find  the  quotient  of  76  divided  by  .2 5. ft     Story. 

Division.     Case  II. 

5.  Divide  $.63  by  7.     This  means    5* . 


(h)  Find  the  quotient  of  $87.15  divided  by  7.     Story. 
(i)  Find  the  quotient  of  $375.50  divided  by  25.     Story. 

(1)  Tell  the  meaning  of  each  of  the  following,  (2)  solve,  and 
(3)  tell  a  suggested  number  story : 

(j)  Divide  $256  by  $8.  (k)  Divide  $256  by  8. 

(1)  Divide  $24.36  by  $.04.       (m)  Divide  $24.36  by  4. 
(n)  Divide  $53.6  by  $.8  (o)  Divide  $53.6  by  8. 

*  These  figures  refer  to  notes  on  pages  8  and  9. 

t  Find  how  many  times  25  hundredths  are  contained  in  4675  hundredths. 

$  Find  how  m»ny  times  5  tenths  are  contained  in  8740  (874.0)  tenths. 

§  Find  how  many  times  25  hundredths  are  contained  in  9600  (96.00)  hundredths. 

II  Find  how  many  times  25  hundredths  are  contained  in  4650  (46.50)  hundredths. 

Tf  Find  how  many  times  25  tenths  are  contained  in  575  (57.5)  tenths. 
**  Find  how  many  times  25  hundredths  are  contained  in  6850  (68.50)  hundredths, 
tt  Find  how  many  times  25  hundredths  are  contained  in  7600  (76.00)  hundredths 


84  COMPLETE    AKITHMETIC. 

DENOMINATE    NUMBERS. 

1.  24  sheets  of  paper  are  1  qvAix. 

2.  20  quires  of  paper  are  1  ream. 

3.  One  ream  of  paper  is  sheets. 

4.  One  half  of  a  ream  is  quires,  or  sheets. 

5.  One  fourth  of  a  ream  is  quires,  or  sheets. 

6.  George  bought  a  ream  of  paper  for  $2  and  sold  it  at 
the  rate  of  2  sheets  for  a  cent ;  he  gained  cents. 

7.  Arthur  bought  a  ream  of  paper  for  $2.25  and  sold  it 
at  15^  a  quire ;  he  gained  cents. 

REVIEW. 

8.  Ten  thousand  feet  are  nearly  miles. 

9.  Fifteen  thousand  feet  are  nearly  miles. 

(a)  Twenty  thousand  feet  are  how  many  feet  less  than 
four  miles  ? 

(b)  Five  miles  are  how  many  feet  more  than  25000  feet? 

10.  One  thousamd  feet  are  nearly  one  th  of  a  mile. 

(c)  Two  fifths  of  a  mile  are  how  many  feet  more  than 
2000  feet  ?     (d)  f  of  a  mile,  than  3000  ft.  ? 

11.  Nine  hundred  feet  are  rods  and  feet. 

12.  One  thousand  feet  are  rods  and  feet. 

13.  One  hundred  feet  are  yards  and  foot. 

14.  Two  hundred  feet  are  yards  and  feet. 

15.  Three  hundred  feet  are  yards. 

(e)    Change  3  mi.  to  ft.  (f)  Change  3  mi.  to  rd. 

(g)    Change  3  A.  to  sq.  rd.  (h)  Change  3  cd.  to  cu.  ft. 

(i)    Change  3|-  mi.  to  ft.  (j)  Change  3|-  mi.  to  rd. 

(k)  Change  3|-  A.  to  sq.  rd.  (1)  Change  3|-  cd.  to  cu.  ft. 

(m)  Change  3^  rd.  to  ft.  (n)  Change  3|  yd.  to  in. 


PARI    I.  85 

MEASUREMENTS. 

1.  In  one  cubic  yard  there  are  cubic  feet. 

(a)  To  dig  a  cellar  15  feet  long,  12  ft.  wide,  and  6  feet 
deep  would  require  the  removal  of  how  many  cubic  feet  of 
earth  ?     (b)  How  many  cubic  yards  ? 

(c)  How  many  square  feet  in  the  floor  of  the  cellar  de- 
scribed in  problem  (a)  ?     (d)  How  many  sq.  yards  ? 

2.  To  make  an  excavation  2  yards  long,  2  yards  wide,  and 
2  yards  deep  would  require  the  removal  of cubic  yards. 

(e)  How  many  cubic  feet  in  the  excavation  described  in 
problem  2  ? 

3.  The  area  of  one  of  the  faces  of  a  2 -yard  cube  is  

square  yards  ;  of  all  the  faces, square  yards. 

(f)  The  area  of  all  the  faces  of  a  2 -yard  cube  is  how  many 
square  feet  ? 

REVIEW. 

4.  A  piece  of  lam)  20  rods  by  8  rods  contains  . 

(g)  How  many  acres  in  a  piece  of  land  20  rd.  by  46  rd.  ? 

5.  A  shed  16  feet  long,  10  feet  wide,  and  6  feet  high 
contains  cubic  feet. 

(h)  How  many  cords  of  wood  will  the  shed  described  in 
problem  5  contain  ? 

Find  the  number  of  acres  in  each  of  the  following : 

(i)  80  rods  by  24|-  rods.  (j)  40  rods  by  73  rods. 

(k)  20  rods  by  52  rods.  (1)  30  rods  by  84  rods. 

Find  the  number  of  cords  in  each  of  the  following : 
(m)  8  ft.  by  12  ft.  by  8  ft.  (n)  16  ft.  by  8  ft.  by  8  ft. 

(o)   16  ft.  by  4  ft.  by  8  ft.  (p)  20  ft.  by  8  ft.  by  8  ft. 


86  COMPLETE   ARITHMETIC. 

RATIO   AND   PROPORTION. 

1.  One  fourth  of  .36  is  .  .36  is  i  of  . 

2.  One  fifth  of  .45  is  .  .45  is  i  of  . 

3.  One  eighth  of  .32  is  .  .32  is  J  of  . 

(a)  Find  1  fourth  of  9.86.*        (b)  9.78  is  ^  of  what  ? 
■  (c)  Find  1  fifth  of  8.24.  (d)  8.24  is  ^  of  what  ? 

4.  Five  eighths  of  .32  are  .        .35  is  |-  of  . 

5.  Three  fifths  of  .45  are  .         .45  is  |  of  . 

(e)  Find  |  of  8.40.  (f)   8.40  is  |  of  what  number  ? 

(g)  Find  I  of  9.45.  (h)  9.45  is  f  of  what  number? 

6.  .32  is of  .40.  .40  is of  .32. 

7.  .24  is of  .40.  .40  is of  .24. 

8.  .24  is of  .56.  .56  is of  .24. 

9.  .40  is —  of  .64.  .64  is of  .40. 

10.  Three  fourths  of  .32  are  1  hal^  of  hundredths. 

(i)  Three  fourths  of  9.36  are  1  half  of  what  number  ? 
(j)   One  half  of  9.36  is  4  fifths  of  what  number  ? 

11.  If  .32  of  an  acre  of  land  is  worth  $20,  .40  of  an  acre 
is  worth  dollars. 

(k)  If  .32  of  an  acre  of  land  is  worth  $24.56,  how  much 
is  .40  of  an  acre  worth  at  the  same  rate  ? 

12.  If  .40  of  a  ton  of  coal  is  worth  $4.00,  .32  of  a  ton  is 
worth  . 

(1)  If  .40  of  a  ton  of  coal  is  worth  $3.25,  how  much  is 
.32  of  a  ton  worth  ? 

*  Think  as  suggested  by  the  following :    J  of  9  is  2  with  1  remainder ;  1  =  10 
tenths  ;  10  tenths  +  8  tenths  are  18  tenths  ;  J  of  18  tenths  is  4  tenths  with 
2  tenths  remainder ;  2  tenths  =  20  hundredths  ;  20  hundredths  +  6  hun-         4)9^ 
dredths  =  26  hundredths  ;   i  of  26  hundredths  =  G   hundredths  with  2  ^•'*^ 

hundredths  remainder ;  2  hundredths  =  20  thousandths  ;  i  of  20  thousandths  =  } 
thousandths. 


PART    I. 


87 


PERCENTAGE. 


30  per  cent  =  .30  =  .3 
70  per  cent  =  .70  =  .7 
90  per  cent  =  .90  =  .9  =  -j%. 


3 

10"- 

7 

r"o- 


(1) 

1.  30%  of    60  = 

2.  40%  of    60  = 

3.  50%  of    60  = 

4.  60%  of    60  = 

5.  70%  of  140  = 

6.  80%  of  160  = 

7.  90%  of  180  = 

(3) 


8.  4  is 

9.  28  is 

10.  5  is 

11.  15  is 

12.  6  is 


%  of  40. 

%  of  40. 

%  of  50. 

%  of  50. 

%  of  60. 


(2) 

60  is  30%  of  — 

60  is  40%  of  — 

60  is  50%  of  — 

60  is  60%  of  — 

140  is  70%  of  . 

160  is  80%  of  — 

180  is  90%  of  — 

(5) 

12  is  %  of  40. 

36  is  %  of  40. 

45  is  %  of  50. 

35  is  %  of  50. 

42  is  %  of  60. 


13.  In  a  certain  orchard  there  were  120  trees;  10%  of 
them  were  pear  trees  and  30%  of  them  were  cherry  trees; 
there  were  pear  trees  and  cherry  trees. 

14.  In  another  orchard  there  were  24  pear  trees  ;  these 

were  30%  of  all  the  trees  in  the  orchard;  there  were  

trees  in  the  orchard. 

15.  There  were  60  trees  in  an  orchard ;  42  of  them  were 

apple  trees  and  the  remainder  pear  trees ;  there  were  

pear  trees  ;  • per  cent  of  the  trees  in  the  orchard  were 

apple  trees  and  per  cent  of  them  were  pear  trees. 

16.  Twenty-five  %  of  the  trees  in  an  orchard  died  and 
were  removed ;  there  remained  36  trees ;  trees  died. 


88  COMPLETE    ARITHMETIC. 

PERCENTAGE. 

1.  30  per  cent  of  42  =  (^V  of  42  is  4.2.) 

2.  90  per  cent  of  51  =  90  per  cent  of  31  = 
(a)  Find  30%  of  275.  (b)  Find  70%  of  362. 
(c)  Find  90%  of  436.  (d)  Find  30%  of  475. 
(e)  Find  3%  of  536.  (f)  Find  7%  of -824. 

(g)  A  lawyer's  commission  for  collecting  money  was  10%; 
he  collected  $954.  How  much  of  the  money  should  he 
keep  ?  and  how  much  should  he  "  pay  over  "  to  the  man  for 
whom  he  collected  the  money  ? 

(^) 

3.  3.2  is  10  per  cent  of  .         4.6  is  10%  of  . 

4.  9.6  is  30  per  cent  of  .  14.7  is  70%  of  . 

(h)  372.3  is  30%  of  what?  (i)  562.1  is  70%  of  what? 
(j)  291.6  is  90%  of  what?  (k)  752.4  is  30%  of  what? 
(1)  279  is  3%  of  what?  (m)  539  is  7%  of  what? 

(n)  A  lawyer's  commission  for  collecting  money  was  10%; 

his  commission  amounted  to  $76.20.  How  much  did  he 
collect  ?  and  how  much  should  he  "  pay  over  "  to  the  man  for 
whom  he  collected  the  money  ? 

(5) 

5.  5.1  is  per  cent  of  51.  8.3  is  %  of  83. 

6.  84  is  per  cent  of  840.  8.40  is  %  of  840. 

7.  7.54  is  .  per  cent  of  75.4.  7.54  is  %  of  754. 

8.  A  lawyer  collected  $754;  he  retained  as  his  commis- 
sion $75.40  of  this  sum  and  paid  the  remainder  to  the  man 
for  whom  he  collected  the  money ;  the  lawyer's  commission 
for  collecting  was  %. 


PART    I.  89 

KEVIEW. 

1.  Six  sheep  cost  $33  ;  the  average  cost  per  head  was . 


(a)  In  a  certain  schoolroom,  on  Monday  there  were  35 
pupils ;  Tuesday,  38  ;  Wednesday,  36  ;  Thursday,  37,  and 
Friday,  39.     What  was  the  average  daily  attendance  ? 

2.  Divide  j%  by  |.     (Change  to  ths.)     Story. 

(b)  Divide  $323|  by  $1|.     (Change  to  ths.)     Story. 

(c)  Divide  $323.75  by  $1.25.     This  means  . 

Compare  the  answers  to  problems  (b)  and  (c).  Why  are  they 
alike? 

3.  In  ^  of  a  ream  of  paper  there  are  quires. 

4.  At  3  sheets  for  2  cents,  1  quire  of  paper  costs  . 


(d)  At  3  sheets  for  2  cents,  how  much  will  one  ream  of 
paper  cost  ? 

5.  In  2  square  yards  there  are  square  feet. 

6.  In  2  cubic  yards  there  are  cubic  feet. 

(e)  In  18  cubic  yards  there  are  how  many  cu.  ft. 

(f)  In  48  square  yards  there  are  how  many  square  feet  ? 

(g)  How  many  square  feet  of  blackboard  in  this  room  ?  * 
(h)  How  many  sq.  yd.  of  blackboard  in  this  room  ?  * 

(i)  How  many  cubic  feet  of  earth  must  be  removed  to 
make  an  excavation  6  ft.  by  9  ft.  by  15  ft.  ?  (j)  How  many 
cubic  yards  ? 

(k)  326  +  432  +  28  +  175  +  326  +  47  +  63  +  82  +  96  = 
(1)  274  +  386  +  27  +  237  +  528  +  35  +  72  +  51  +  46  = 
(m)  438  +  275  +  43  +  324  +  136  +  32  +  85  +  37  +  91  = 
(n)  243  +  139  +  18  +  914  +  326  +  12  +  72  +  36  +  75  = 

*  Require  an  approximate  answer  only. 


90  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    REVIEW. 

1.  Two  boys  start  from  two  cities  that  are  67  miles 
apart  and  ride  toward  each  other;  one  rides  8  miles  an 
hour;  the  other  rides  12  miles  an  hour;  in  one  hour  they 
ride  miles ;  in  3  hours  they  are  •  miles  apart. 

(a)  Two  trains  start  from  two  cities  that  are  1000  miles 
apart  and  move  toward  each  other;  one  goes  25  miles  an 
hour;  the  other  goes  30  miles  an  hour.  At  the  end  of  12 
hours  how  far  are  the  trains  apart  ? 

2.  A  man  bought  land  for  $80  and  sold  it  for  $88 ;  he 
gained  $2  an  acre ;  there  were  acres. 

(b)  A  man  bought  land  for  $1850,  and  sold  it  for  $2220, 
th(}reby  gaining  $5  an  acre.  How  many  acres  did  he  buy  ? 
(c)  How  much  did  he  pay  per  acre  for  it  ? 

3.  A  boy  earned  some  money,  spent  f  of  it,  and  had  $6 
left ;  he  spent  dollars. 

(d)  A  man  earned  some  money,  spent  |  of  it,  and  had 
$25.20  left.     How  much  did  he  spend  ? 

4.  The  product  of  two  numbers  is  28  ;  one  of  the  num- 
bers is  7  ;  the  other  number  is  . 

(e)  The  product  of  two  numbers  is  240  ;  one  of  the  num- 
bers is  15.     What  is  the  other  number  ? 

5.  Fifty  per  cent  of  my  money  is  $12,  and  John  has  3 
times  as  much  as  I  have ;  John  has  dollars. 

(f)  Fifty  per  cent  of  Mr.  A's  money  is  $36.45  and  Mr. 
B  has  3  times  as  much  as  Mr.  A.     Mr.  B  has . 

6.  I  have  $24;  50%  of  my  money  equals  33-i^  per  cent 
of  Bernie's  money ;  Bernie  has  dollars. 

(g)  Mr.  C  has  $72.30;  50%  of  Mr.  C's  money  equals 
33^%  of  Mr.  D's  money.     How  much  money  has  Mr.  D  ? 


PART   I. 

SIMPLE    NUMBERS. 

Niimher  of  poitnds  of  milk  received. 


'91 


PATRONS. 

M. 

236 
149 
375 

97 
201 
385 

87 
230 
155 
435 

T. 

224 

167 
384 
84 
196 
397 
146 
231 
160 
424 

W. 

259 
134 
363 
93 
205 
388 
175 
229 
145 
412 

T. 

263 
158 
356 

86 
208 
394 

82 
233 
170 
456 

F. 

248 
175 
325 
92 
199 
396 
185 
228 
180 
428 

S. 

227 
136 
364 

87 
215 
399 

96 
229 
175 
435 

S. 

23G 
145 
356 
88 
218 
401 
133 
235 
194 
457 

am't. 

L.  R.  Orr 

Levi  Smith 

Henry  Judd 

Peter  Johnson 

Wm.  Jones 

E.C.Ford 

{^) 

(b) 

(0) 

(d) 
(e) 
(f) 
(g) 
i^) 
(i) 
(J) 

Otto  Fisk 

Fay  Winslow 

Albert  Davis 

Jno.  Harris 

Total 

(k) 

(1) 

(m) 

(n) 

(o) 

(P) 

(q) 

(!•)       (s) 

The  above  is  a  statement  of  the  milk  received  at  a  small  creamery 
for  one  week,  (a)  to  (j)  Find  the  amount  delivered  by  each  patron, 
(k)  to  (q)  Find  the  amount  received  at  the  creamery  each  day. 
(r)  Find  the  sum  of  (k)  to  (q)  inclusive,  (s)  Find  the  sum  of  (a) 
to  (j)  inclusive. 

(t)  Find    tlie    average    amount    received    daily    at    the 
creamery. 


EEVIEW. 


etc. 


1.  Common  multiples  of  15  and  10  are  ,  - 

The  least  common  multiple  of  15  and  10  is  . 

2.  The  prime  factors  of  75  are  ,  ,  and  . 

(;u)  What  are  the  prime  factors  of  2750  ?      (v)  Of  280  ; 

3.  3,  3,  and  7  are  the  prime  factors  of  .    Sixty-three 

is  exactly  divisible  by  3  ;  by  ;  by  ;  and  by  . 


(w) 
27  cu.  ft.)1296  cii.  ft. 


27)1296  cu.  ft. 


92  •  COMPLETE    AKITHMETIC. 


COMMON   FRACTIONS. 

1.  Add  yig  and  -^-^.     The  1.  c.  m.  of  15  and  10  is 


(a)  Find  the  sum  of  5S4.j\,  257^V  225i  and  324i. 

2.  From  8^1^  subtract  5^\.       1-L  =  37.       tV  =  3 o- 

(b)  Find  the  difference  of  7433?^  and  ISl^V 

3.  Multiply  -^\  by  8.     This  means . 

(c)  Find  the  product  of  235yV  multiplied  by  8. 

4.  Multiply  If  by  i.     This  means . 

(d)  Find  the  product  of  724f  multiplied  by  J.     Story, 

6.  Multiply  16  by  |.     This  means . 

(e)  Find  the  product  of  721  multiplied  by  |. 

6.  Multiply  16  by  2|.     This  means . 


(f)  Find  the  product  of  721  multiplied  by  7|. 
7.  Multiply  18|-  by  2i.     This  means 


(g)  Find  the  product  of  432 1-  multiplied  by  4i 

8.  Divide  7  by  |.      (Change  7  to  ths.) 

(h)  Find  the  quotient  of  94  divided  by  |.     Story. 

9.  Divide  -^\  by  yV      (Change  to  ths.) 

(i)  Find  the  qiiotieiit  of  4^i  divided  by  J^.     Story. 

10.  Divide  6f  by  2  J.     (Change  to  ths.) 

(j)    Find  the  quotient  of  74f  divided  by  3^. 

11.  Divide  1|  (V)  by  3.     This  means . 

(k)  Find  the  quotient  of  7|  divided  by  3. 

(1)  (m)             (n)                   (0)  (p) 

Add.  Subtract.  Multiply.           Divide.  Divide. 

375||  436J            753|  3^  yd.)512  yd.  5)636f  yd. 

286yV  182y3_-           24  " 


PART   I.  93 

DECIMAL    FRACTIONS. 

Division.     Case  I. 

1.  Divide  .48  by  .12.     This  means . 

(a)  Find  the  quotient  of  54.24  divided  by  .12.     Story. 

2.  Divide  6  by  .4.     This  means  . 

(b)  Find  the  quotient  of  736  divided  by  .4.     Story. 

3.  Divide  3  by  .04.     This  means . 

(c)  Find  the  quotient  of  78  divided  by  .12.     Story. 

4.  Divide  4.8  by  .12.     This  means . 

(d)  Find  the  quotient  of  40.8  divided  by  .12.     Story. 

(e)  Find  the  quotient  of  43.2  divided  by  1.2.     Story. 

(f)  Find  the  quotient  of  51.6  divided  by  .12.     Story 

(g)  Find  the  quotient  of  75  divided  by  .12.     Story. 

Division.     Case  II. 

5.  Divide  $.56  by  8.     This  means . 


(h)  Find  the  quotient  of  $98.40  divided  by  8.     Story. 
(i)  Find  the  quotient  of  $436.80  divided  by  12.     Story. 

(1)  Tell  the  meaning  of  each  of  the  following,  (2)  solve,  and  (3) 
tell  a  suggested  number  story. 

(j)  Divide  $924  by  $7.  (k)  Divide  $924  by  7. 

(1)  Divide  $29.34  by  $.06.         (m)  Divide  $29.34  by  6. 
(n)  Divide  $55.3  by  $.7.  (o)  Divide  $55.3  by  7. 

(p)  Multiply  $265  by  .3.  (q)  Multiply  $265  by  2.3. 

(r)  Multiply  $265  by  .03.  (s)  Multiply  $265  by  .23. 

(t)  Multiply  $265  by  4.23. 

(u)At  $.12  a  dozen,  how  many  dozen  eggs  can  I  buy  for 
$63.84? 


94 


COMPLETE    ARITHMETIC. 


DENOMINATE    NUMBERS. 


(a)  Copy,  complete,  and  receipt  the  following  Bill : 

Waukegan,  III.,  July  30,  1896. 
Mr.  Kichard  Yates, 

Bought  of  C.  F.  Brown  &  Co. 


July 

1 

a 

1 

i( 

5 

(( 

8 

<c 

8 

« 

19 

15     lb.  Sugar, 
5     lb.  Raisins, 
2 1-  lb.  Cheese, 
1     sack  Flour, 
^    bu.  Potatoes, 
5    lb.  Coffee, 


$.05 
.07 
.16 

$.60 
.35 


Eeceived  Payment, 


(a) 


25 


REVIEW. 

1.  500  sheets  of  paper  are  1  ream  and  sheets. 

2.  500  feet  are  rods  and  feet. 

3.  500  rods  are  1  mile  and  rods. 

4.  500  sq.  rd.  are  3  acres  and  sq.  rd. 

5.  500  inches  are  feet  and  inches. 

6.  500  feet  are  yards  and  feet. 

7.  500  pounds  are of  a  ton. 

8.  When  hay  is  $8  a  ton,  5000  lb.  cost  dollars. 

(b)  Load  of  straw,  gross  weight,  3380  lb.;  tare,  1580  lb. 
Find  the  value  at  $5.50  a  ton. 

(c)  Multiply  40  sq.  rods  by  24  and  divide  the  product  by 
160  square  rods.     Stor?/. 

(d)  Multiply  24  cubic  feet  by  8  and  the  product  thus  ob- 
tained by  6.     Divide  the  last  product  by  128  cu.  ft.     Story. 


PART   I.  95 

MEASUREMENTS. 


3^  in. 

.s 

This  diagram  represents  a  piece  of  land. 

-h|n 

It  is  drawn  on  a  scale  of  |  inch  to  the  rod. 

CM 

1.  The  land  represented  by  the  above  diagram  is  

rods  long.     It  is  rods  wide. 

(a)  Find  the  perimeter  of  the  piece  of  land  represented  by 
the  diagram,  (b)  Find  its  area  in  square  rods,  (c)  Find  its 
area  in  acres,  (d)  How  much  is  the  land  worth  at  S40  an 
acre  ? 

2.  Draw  carefully  upon  your  slate  or  paper,  on  a  scale  of 
J  in.  to  the  rod,  a  diagram  of  a  rectangular  piece  of  land 
that  is  30  rods  by  36  rods. 

(e)  Find  the  perimeter  of  the  land  represented  by  the 
diagram  you  haVe  drawn,  (f)  Find  its  area  in  square  rods, 
(g)  Find  its  area  in  acres.  (h)  How  much  is  the  land 
worth  at  $60  an  acre  ?  (i)  How  much  will  it  cost  to  build 
a  fence  around  it  at  50^  a  rod?  (j)  Find  its  area  in  square 
yards. 


96  COMPLETE    ARITHMETIC. 

RATIO   AND   PROPORTION. 

1.  One  fifth  of  2  (2.0)  is  *  ^  of  3  is  . 

2.  One  fourth  of  6  (6.0)  is .  i-  of  4  is  . 

3.  One  eighth  of  4  (4.0)  is  .  ^  of  2  is  . 

(a)  Find  1  fourth  of  70.  (b)  Find  \  of  79. 

(c)  Find  1  fifth  of  83.  (d)  Find  -i-  of  66. 

4.  1  is of  27.  2  is of  27. 

5.  3  is of  27,  or  — .f         4  is of  27. 

6.  5  is of  27.  ■    6  is of  27,  or  — . 

7.  7  is of  27.  8  is of  27. 

8.  9  is of  27,  or  — .  10  is  —  —  of  27. 

9.  11  is of  27.  12  is of  27,  or  -  -. 

10.  13  is of  27.  14  is of  27. 

11.  15  is of  27,  or  — .       16  is  —  —  of  27. 

12.  17  is of  27.  18  is of  27,  or  — . 

13.  1.8  is of  2.7.  2.7  is of  1.8. 

14.  .18  is of  .27.  .27  is of  .18. 

15.  .19  is of  .27.  .27  is  ■  of  .19. 

16.  1.9  is of  2.7.  2.7  is of  1.9. 

(e)    120  is  what  part  of  140  ?  %  (f)    140  of  120  ? 

(g)   175  is  what  part  of  185  ?  (h)  185  of  175  ? 

(i)    120  is  what  part  of  128  ?  (j)   128  of  120  ? 

(k)  120  is  what  part  of  160  ?  (1)   160  of  120  ? 

(m)  24  is  what  part  of  144  ?  (n)  48  of  144  ? 

(o)   96  is  what  part  of  144  ?  (p)  100  of  144  ? 

(q)  120  is  what  part  of  150  ?  (r)    125  of  150  ? 

*  J  of  20  tenths. 

1 3  is  three  twenty-sevenths  of  27,  or  ^  of  27. 

X  In  each  such  problem  as  (e) ,  the  number  that  follows  the  word  of  always 
becomes  the  denominator.    120  is  \l%  =  (}  =  $  of  140. 


PART    I. 


97 


PEKCENTAGE. 


1.  One  %  of  $435  =• 

2.  One  %  of  S832  = 

3.  One  %  of  $38.4  = 

4.  One  %  of  $56.30  = 

5.  One  %  of  $8324  = 


{1) 


(a)  Find  7%  of  $435. 

(b)  Find  9%  of  $832. 

(c)  Find  8%  of  $38.4. 

(d)  Find  6%  of  $56.30, 

(e)  Find  5%  of  $8324. 


{2) 


6.  63  is  7%  (7  hundredths)  of  . 

(f)  364  is  7%  of  what?        (g)  635  is  5%  of  wha/. ? 

7.  72  is  9%  (9  hundredths)  of  . 

(h)  657  is  9%  of  what  ?        (i)  584  is  8%  of  whnt  ? 

8.  54  is  6%  (6  hundredths)  of  . 

(j)  456  is  6%  of  what  ?        (k)  504  is  7%  ot  v/hat  ? 


(-5) 


is  %  of  $200.* 

is  what  %  of  $200  ? 
is  %  of  $300. 


9.  $12 

(1)  $54 
11.  $21 
(n)  $84 
13.  $36 
(p)  $96 
15.  $40 
(r)    $95  is  what  %  of  $500  ? 


is  what  %  of  $300  ? 

is  %  of  $400. 

is  what  %  of  $400  ? 
is  %  of  $500. 


10.  $13  is  -^.-%  of  $200 
(m)  $45  is  what  %  of  $200  ' 

12.  $22  is  ^ %  of  $300 

(o)  $85  is  what  %  of  $300  ' 

14.  $38  is  %  of  $400 

(q)  $98  is  what  %  of  $400  ' 

16.  $42  is  %  of  $500, 

(s)   $96  is  what  %  of  $500  ? 


*Lead  the  pupil  to  observe  that  he  has  already  learned  to  solve  "  Case  3  "  prob- 
lems in  percentage  in  two  ways.  Sometimes  he  has  found  what  part  one  number 
is  of  another  and  has  changed  the  fraction  thus  obtained  to  hundredths.  See  prob- 
lems 11  to  17,  page  57.  Sometimes  he  has  found  one  per  cent  of  the  base  and  has 
used  this  number  as  a  divisor  to  obtain  the  required  result.  See  page  58,  problem 
(o);  iiage  68,  problems  3  to  5 ;  page  78,  problems  4  to  6.  The  "  Case  3  "  problems  on 
this  page  and  the  greater  number  of  those  that  follow  in  Part  I.,  can  be  most  readily 
solved  by  the  second  method.  12  is  what  5^  of  200?  One  %  of  200  is  2;  therefore  12  is  6 
per  cent  qf200;  13  is  6i  per  cent;  6I4.  is  27  per  cent,  and  U5  is  22\  per  cevi. 


98  COMPLETE   ARITHMETIC. 

PERCENTAGE. 

1.  Three  %  of  $620  =  (a)  Find  9%  of  $620. 

2.  Eighteen  is  3%  of  .      (b)  $75  is  3%  of  what? 

3.  Eighteen  is %  of  600.  (c)  138  is  what  %  of  600? 

4.  Eight  %  of  $220  -  (d)  Find  8%  of  $738. 

5.  Forty-eight  is  8%  of .  (e)  $76  is  4%  of  what? 

6.  Thirty-six  is %  of  600.  (f)  174  is  what  %  of  600? 

(g)  A  farmer  had  825  sheep;  he  sold  8%  of  them.    How 

many  sheep  did  he  sell  ?  (h)  How  many  sheep  did  he  have 
left? 

(i)  A  dealer  bought  some  sheep  and  immediately  sold  72 
of  them ;  these  were  8  %  of  all  he  purchased.  How  many 
sheep  did  he  purchase  ?  (j)  How  many  sheep  did  he  have 
left? 

(k)  Mr.  B  had  400  sheep;  he  sold  48  of  them.  What  % 
of  his  sheep  did  he  sell  ? 

(1)  A  man  had  $485 ;  he  spent  7%  of  his  money.  How 
much  money  did  he  spend  ?  (m)  How  much  money  did  he 
have  left? 

(n)  Peter  Gray  paid  $35.50  for  a  suit  of  clothes ;  this  was 
0  %  of  all  he  earned  in  a  year.  How  much  did  he  earn  in 
a  year? 

(o)  Judge  Branson  had  $600  in  the  bank;  he  drew  out 
$198.     What  %  of  his  deposit  did  he  draw  out  ? 

7.  A  lawyer  collected  $800  ;  he  retained  as  his  commis- 
sion $56  of  this  sum  and  paid  the  remainder,  dollars, 

to  the  man  for  whom  he  collected  the  money ;  the  lawyer's 
commission  for  collecting  was  %. 

(p)  A  lawyer  collected  $850;  he  retained  as  his  commis- 
sion for  collecting  $51  of  this  sum.  What  per  cent  did  the 
lawyer  charge  for  making  the  collection  ? 


PART    I.  ,99 

REVIEW. 

(a)  A  creamery  received  milk  as  follows :  Monday,  8250 
lb. ;  Tuesday,  7640  lb. ;  Wednesday,  8375  lb. ;  Thursday,  7230 
lb.;  Friday,  6953  lb.;  Saturday,  8424  lb. ;  Sunday,  7651  lb. 
Find  the  number  of  pounds  received  during  the  week,  (b) 
Find  the  average  daily  receipts. 

1.  Add  ^^  and  y^^.      (Change  to  ths.) 

(c)  Find  the  sum  of  375||,  246|,  82l3i_,  and  234. 

(d)  Find  the  product  of  372  multiplied  by  6|. 

(e)  Find  the  product  of  372  multiplied  by  6.75. 
Compare  the  answers  to  problems  (d)  and  (e). 

(f)  Make  a  bill  containing  the  following  items,  and  find 
the  amount : 

Aug.  2,  24  lb.  sugar  @  5^;  Aug.  5,  ^  lb.  tea  @  70^; 
Aug.  10,  2  gal.  syrup  @  35^ ;  Aug.  13,  5  bu.  po- 
tatoes @  55^;  Aug.  18,  5  cans  peaches  @  18^; 
Aug.  18,  5  gal.  kerosene  @  15^. 

2.  Draw  a  diagram  of  a  piece  of  land  40  rods  by  36  rods, 

scale,  ^  in.  to  a  rod.     The  diagram  is  inches  long  and 

inches  wide. 

(g)  How  many  square  rods  in  the  piece  of  land  described 
in  problem  2  ?  (h)  How  many  acres  ?  (i)  The  perimeter  of 
the  field  is  how  many  rods  ? 

3.  3  is of  25.     3  twenty-fifths  are  —  hundredths. 

4.  4  is of  25.     4  twenty-fifths  are  —  hundredths. 

(j)   85  is  what  part  of  95  ?  (k)  95  of  85  ? 

(1)    95  is  what  part  of  105  ?  (m)  105  of  95  ? 

(n)  105  is  what  part  of  115  ?        (o)   115  of  105  ? 


100.  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    PROBLEMS. 

1.  Horace  spent  |-  of  his  money  and  had  24^  left:  he 

spent  — — •  cents.     Before  spending  any  money  he  had 

cents. 

(a)  Mr.  Green  spent  |  of  his  money  and  had  $56.48  left. 
How  much  money  did  he  spend  ?  (b)  Before  spending  any 
money  how  much  did  he  have  ? 

2.  At  2|^^  an  ounce,  |-  of  a  pound  of  cinnamon  costs 
cents. 


(c)  Find  the  cost  of  5^  pounds  of  cloves  at  1^^  per  ounce. 

3.  A  farmer  mixed  10  bushels  of  oats  with  15  bushels 

of  corn ;  together  there  were  bushels  of  grain.     What 

part  of  the  mixture  was  oats  ?      What  part  was  corn  ? 

(d)  A  dealer  mixed  250  bushels  of  oats  with  350  bushels 
of  corn.  What  part  of  the  mixture  was  oats  ?  (e)  What 
part  was  corn  ? 

4.  Eddie,  Willie,  and  Fay  divided  a  melon  among  them- 
selves. Eddie  had  1  fourth  of  the  melon.  Will  had  ^  of 
what  remained  after  Eddie  had  taken  his  share.  What 
part  of  the  melon  did  Will  have  ?  What  part  of  the  melon 
was  left  for  Fay  ? 

(f)  An  estate  was  divided  as  follows  :  The  widow  received 
1  third  of  it,  and  each  of  five  children  received  1  fiftli  of 
what  remained  after  the  widow  had  received  her  share. 
What  part  of  the  whole  estate  did  each  child  receive  ? 

5.  A  had  3  sevenths  of  an  estate' and  B  had  4  sevenths 
of  it.     A's  share  was  equal  to  what  part  of  B's  share  ? 

(g)  If  the  estate  mentioned  in  problem  5  was  wor^b 
$9660,  what  was  the  value  of  A's  share  ?    (h)  Of  B's  share  ? 


PART    I, 


1*01 


SIMLPE    NUMBERS.'- 
Time-booh^  showing  hours  worked  for  one  week* 


Name. 

Daily 
Wages 

M. 

T. 

W. 

T. 

F. 

S. 

Hours 

Days 

Am't 
earn'd 

John  Miller 

T.Benjamin 

D.  Gordon 

N.  Hanson 

A.  Gillett 

Wm.  Price 

R.L.  Wing 

J.  Carlson 

S.  Spencer 

S.  L.  Judd 

$2.00 
2.50 
3.00 
2.00 
2.40 
3.50 
2.60 
2.60 
3.00 
2.20 

10 

10 

12 

9 

10 
10 
10 
10 
11 
10 

10 

9 

12 

10 

10 

12 

9 

8 

10 
12 

8 
10 
12 
12 
10 
12 
9 
0 
12 
10 

9 
8 

10 
11 
10 
12 
10 
10 
10 
0 

12 
10 
12 
10 
10 
10 
12 
12 
10 
0 

10 
15 
12 
9 
10 
11 
12 
12 
12 
15 

(a) 
(b) 
(c) 
(d) 

(i) 

(i) 

(k) 

(1) 
(m) 

(n) 
(o) 
(P) 
(q) 

u 

(t) 

(u) 

(V) 

(w) 

(X) 

(y) 

(z) 

(aa) 

(bb) 

(cc) 

(dd) 

Total  hours 

(ee) 

(ff) 

(gg) 

(hh) 

(ii) 

ij) 

(kk) 

(11) 

(mm) 

The  above  is  from  the  time-book  of  a  small  manufactory. 

(a  to  j)  Find  the  number  of  hours  worked  by  each 
employe.     ' 

(k  to  t)  Find  the  number  of  days  (10  hours)  worked  by 
each  employe. 

(u  to  dd)  Find  the  amount  earned  by  each  employe. 

(ee  to  jj)  Find  the  number  of  hours  of  work  done  each 
day. 

(kk)  Find  the  sum  of  (a)  to  (j)  inclusive ;  of  (ee)  to  (jj) 
inclusive. 

(11)  Find  the  sum  of  (k)  to  (t)  inclusive. 

(mm)  Find  the  sum  of  (u)  to  (dd)  inclusive. 


-,  etc. 


KEVIEW. 

1.  Common  multiples  of  15  and  20  are  , 

The  least  common  multiple  of  15  and  20  is  . 

*Copy  this  time-book  and  write  numbers  in  place  of  the  letters  ;  that  is,  instead 
of  (a),  write  the  number  of  hours  worked  by  John  Miller ;  in  place  of  (k),  write  the 
nxunber  of  days  worked  by  John  Miller,  etc.    Call  8  hours  a  day's  work. 


102  COMPLETE    ARITHMETIC. 


•COMMON    FRACTIONS. 

1.  Add  I,  yV  and  f     The  1.  c.  m.  of  8, 10,  and  5 
^a)  Eind  the  sum  of  896|,  507^3_^  and  344|. 

2.  From  9|-  subtract  3.2.     1^  =  ^  .2=^. 
(b)  Find  the  difference  of  1754f  and  592.7. 

3.  Multiply  2y\-  by  5.     This  means 


IS 


(c)  Find  the  product  of  757y\-  multiplied  by  5. 

4.  Multiply  1|-  by  i      This  means  . 

(d)  Find  the  product  of  536|-  multiplied  by  i      ^^^ry. 

5.  Multiply  21  by  |.     This  means  . 

(e)  Find  the  product  of  681  multiplied  by  |. 

6.  Multiply  21  by  2f.     This  means  . 


(f)  Find  the  product  of  681  multiplied  by  8|. 
7.  Multiply  11 1-  by  2  J.     This  means  — 


(g)  Find  the  product  of  273|-  multiplied  by  4^. 

8.  Divide  7^  by  |.      (Change  7}  to  ths.) 

(h)  Find  the  quotient  of  97|-  divided  by  |.     Story. 

9.  Divide  |  by  y3_.      (Change  to  ths.) 

(i)  Find  the  quotient  of  7-|  divided  by  -f^.     Story. 

10.  Divide  7f  by  2i       (Change  to  ths.) 

(j)   Find  the  quotient  of  86|-  divided  by  S^. 

11.  Divide  2|  (Y)  by  3.     This  means  , 

(k)  Find  the  quotient  of  8|  divided  by  9. 

12.  Divide  21f  by  5.     This  means 


(1)  Find  the  quotient  of  62 IJ  divided  by  5. 

(m)  (n)               (o)                      (p)                         (q) 

Add.  Subtract.    Multiply.            Divide.  Divide. 

478|-  539|-          738|          3J  ft.)720i- ft.  3)745 f  ft. 


896f         182.3  28 


PART    I. 


103 


DECIMAL    FRACTIONS. 


Case  I. 

This  means  . 

(b)  47.275-1- .025=     Story. 

This  means . 

(d)  375.6  -i-  .05  =      Story.'' 

This  means  . 

(f)  256.4 -^.004=       Story. 
(h). 35  ^.004=       Story. 
(j)  .35  ^  .025  =       Story. 

Case  II. 
This  means  . 


Division. 

1.  Divide  .072  by  .012. 
(a)  65*088 -f- .012  = 

2.  Divide  6  by  .05. 
(c)  468  ^  .05  = 

3.  Divide  8  by  .004. 
(e)  75  -^  .004  = 
(g)  38.72  ^  .004  = 
(i)   73-^.025  = 

Division. 

4.  Divide  $.066  by  8.  • 
(k)  $24,368  -4-  8  =  (1)  $35,056  ^  8  =       Story. 

(1)  Tell  the  meaning  of  each  of  the  following,  (2)  solve,  and  (3) 
tell  a  suggested  number  story. 

(m)  $336.6  -J-  $6  =        (n)  $336.6  -4-  6  = 

(o)  $336.6  -^  $.6  =        (p)  $336,624  -^  6  = 

(q)  $356  X  .7  =  (r)  $356  x  2.7  =  ' 

(s)  At  $.25  a  dozen,  how  many  lemons  can  I  buy  for 

$36.75  ? 

(t)  I  paid  $36.75  for  25  barrels  of  apples.     How  much 

did  the  apples  cost  per  barrel  ? 

(u)  If  one  acre  of  land  is  worth  $36.75,  how  much  are 

5.6  acres  worth  ? 

(v)  If  2  chairs  cost  $37.24,  how  much  will  9  chairs  cost 

at  the  same  rate  ? 

*  Pupils  will  most  readily  give  a  number  story  in  such  work  as  this,  by  letting 
the  figures  represent  units  of  money,  thus  :  At  $.05  each,  for  $375.6  I  could  buy  7512 
tablets.  When  the  pupils  are  well  grounded  tn  this  work,  they  may  be  encouraged  to 
seek  variety  in  their  number  stories. 


104  COMPLETE    ARITHMETIC. 


DENOMINATE    NUMBERS. 


1.  Twelve  dozen  are  one  gross. 

2.  At  48^  a  gross,  1  dozen  buttons  cost  cents. 

3.  At  5^  a  dozen,  1  gross  of  buttons  cost cents. 

4.  At  50^  a  gross,  6  dozen  buttons  cost  cents. 

5.  At  36^  a  gross,  3  dozen  buttons  cost  cents. 

6.  George  bought  pens  at  50^  a  gross  and  sold  them  at 
the  rate  of  2  for  1  cent ;  on  1  gross  he  gained  . 

(a)  Henry  bought  pens  at  60^  a  gross  and  sold  them  at 
the  rate  of  3  pens  for  2  cents.  How  much  did  he  gain  on 
one  gross  ?     (b)  How  much  on  7  gross  ? 

REVIEW. 

(c)  Make  a  receipted  bill  of  the  following  goods  sold  by 
yourself  to  Mr.  Frank  H.  Armstrong: 

Aug.  2,  15  lb.  shingle  nails  @  4^;  Aug.  10,  6  lb. 
fence  staples  @  5^';  Aug.  20,  130  lb.  No.  9  fence 
wire  @  2|^^;  Aug.  25,  1  keg  20d.  nails  @  $3.25.* 

7.  One  fourth  of  a  ream  of  paper  is  sheets. 

8.  One  fourth  of  a  ton  of  coal  is  pounds. 

9.  One  fourth  of  a  mile  rods. 

10.  When  coal  is  $6  per  ton,  7000  lb.  cost  dollars. 

(d)  Load  of  bran;  gross  weight,  2850  lb.;  tare,  1050  lb. 
Find  the  value  at  $7.50  a  ton. 

(e)  Multiply  36  square  feet  by  24,  and  divide  the  product 
thus  obtained  by  9  square  feet.     Story. 

(f)  Multiply  12  cubic  feet  by  15,  and  the  product  thus 
obtained  by  6.  Divide  the  last  product  by  27  cubic  feet. 
Story. 

*  Ask  the  pupil  to  bring  to  school  a  shingle  nail,  a  fence  staple,  a  piece  of  No.  9 
wire,  and  a  JOd.  nail ;  also  to  learu  the  current  price  of  the  articles  named. 


PART    I.  105 

MEASUREMENTS. 

1.  Draw  carefully  upon  your  slate  or  paper,  on  a  scale 
of  ^  inch  to  the  rod,  a  diagram  of  a  rectangular  piece  of 
land  that  is  30  rods  by  40  rods. 

(a)  Find  the  perimeter  of  the  land  represented  by  the 
diagram  you  have  drawn. 

(b)  Find  its  area  in  square  rods. 

(c)  Find  its  area  in  acres. 

(d)  How  much  is  the  land  worth  at  $72.50  an  acre  ? 

(e)  How  much  will  it  cost  to  build  a  fence  around  it  at 
45^  a  rod  ? 

(f)  A  strip  4  rods  wide  across  one  side  of  the  field  con- 
tains how  many  square  rods  ? 

(g)  A  strip  8  rods  wide  across  one  side  of  the  field  con- 
tains how  many  acres  ? 

2.  Land  5  rods  by  40  rods  contains  . 

3.  Land  7  rods  by  40  rods  contains  . 

4.  Land  9  rods  by  40  rods  contains  . 

Find  the  number  of  acres  in  each  of  the  following : 

(h)  32  rods  by  5  rods.  (i)  32  rods  by  26  rods. 

(j)  32  rods  by  22  rods.  (k)  32  rods  by  34  rods. 

Find  the  number  of  cords  in  each  of  the  following : 

(1)  12  ft.  by  8  ft.  by  4  ft.    (m)  16  ft.  by  6  ft.  by  4  ft. 

(n)  20  ft.  by  4  ft.  by  4  ft.    (o)  21  ft.  by  8  ft.  by  4  ft. 

Find  the  number  of  square  yards  in  each  of  the  following : 
(p)  30  feet  by  40  feet.  (q)  25  feet  by  33  feet, 

(r)  18  feet  by  21  feet.  (s)  35  feet  by  40  feet. 

Find  the  number  of  cubic  yards  in  each  of  the  following : 

(t)  12  ft.  by  6  ft.  by  9  ft.        (u)  15  ft.  by  12  ft.  by  6  ft. 

(v)  15  ft.  by  9  ft.  by  12  ft.     (w)  21  ft.  by  6  ft.  by  18  ft 


106  COMPLETE    ARITHMETIC. 


RATIO   AND    PROPORTION 


r.  One  inch  is of  a  yard. 

2.  Two  inches  are  of  a  yard. 

3.  Three  inches  are  of  a  yard. 

4.  Four  inches  are of  a  yard. 

5.  One  foot  and  2  inches  are  of  a  yard. 

6.  One  foot  and  8  inches  are of  a  yard. 

7.  One  square  rod  is of  an  acre. 

8.  Two  square  rods  are of  an  acre. 

(a)  Twelve  square  rods  are  what  part  of  an  acre  ? 

(b)  Twenty-eight  square  rods  are  what  part  of  an  acre  ? 

(c)  Thirty-six  square  rods  are  what  part  of  an  acre  ? 

9.  100  square  rods  are of  one  acre.     1  acre  is 

of  100  square  rods,  or  and times 

100  square  rods. 

10.  Two  thirds  of  1  ft.  6  in.  are  3  fourths  of  inches. 

(d)  Two  thirds  of  6  ft.  9  in.  are  3  fourths  of  how  many  in.? 

(e)  Two  thirds  of  96  sq.  rd.  are  1  half  of  how  many  sq.  rd.? 

11.  If  36  sq.  rd.  of  land  is  worth  S60,  48  sq.  rd.  is  worth 
dollars. 

(f)  If  36  sq.  rd.  of  land  is  worth  $75.48,  how  much  is  48 
sq.  rd.  worth  at  the  same  rate  ? 

12.  If  2  ft.  3  in.  of  silver  wire  is  worth  24^',  1  yard  is 
worth  cents. 

(g)  If  2  ft.  3  in.  of  platinum  wire  is  worth  $1.65,  how 
much  is  1  yd.  worth  at  the  same  rate  ? 

13.  If  4  sq.  rd.  of  land  is  worth  $5,  1  acre  is  worth 

dollars. 

(ii)  If  4  sq.  rd.  of  land  is  worth  $7.25,  how  much  is  1 
acre  worth  at  the  same  rate  ? 


PAKT    I.  107 

PERCENTAGE. 

1.  One  %  of  640  =  2.  H%  of  640  = 

3.  One  %  of  420  =  4.  2^%  of  420  = 

5.  One  %  of  $972  =  (a)  Find  7|-%  of  $972. 

6.  One  %  of  $876  =  (b)  Find  9^%  of  $876. 

7.  One  %  of  $46.80  =  (c)  Find  8^%  of  $46.80. 

(2) 

8.  84  is  7%  (7  hundredths)  of  . 


(d)  595  is  7%  of  what?  (e)  876  is  6%  of  what  ? 

9.  63  is  9%  (9  hundredths)  of  . 

(f)  747  is  9%  of  what  ?  (g)  737  is  11%  of  what  ? 

10.  72  is  12%  (12  hundredths)  of  . 

(h)  572  is  11%  of  what?        (i)  560  is  10%  of  what?* 

(5) 

11.  18  is  %  of  300.  12.  19  is  %  of  300. 

(j)   87  is  what  %  of  300  ?  (k)  88  is  what  %  of  300? 

13.  24  is  %  of  400.  14.  26  is  %  of  400. 

(1)    196  is  what  %  of  400  ?  (m)  380  is  what  %  of  400? 

15.  42  is  what  %  of  600  ?  16.  46  is  %  of  600. 

(n)   384  is  what  %  of  600  ?  (o)  576  is  what  %  of  600? 

17.  Two  %  of  430  =  (p)  Find  9%  of  430. 

18.  Twenty  is  4%  of  .      (q)  96  is  4%  of  what? 

19.  Thirty  is %  of  600.     (r)  276  is  what  %  of  600  ? 


20.  Twenty-five  per  cent  of  24  is  what  %  of 

21.  Seventy-five  per  cent  of  24  is  what  %  of 


200? 
300? 


*The  pupil  should  see  that  he  may  solve  this  problem  as  he  solved  problem  (h), 
or  he  may  solve  it  by  making  use  of  the  fact  that  lO^i  =  1  tenth. 


108  COMPLETE    ARITHMETIC. 


PERCENTAGE. 


1.  A  nursery-man  set  out  200  trees;  8%  of  them  died; 
-  trees  were  dead  and  trees  were  alive. 


(a)  A  nursery-man  set  out  650  trees;  8%  of  them  died. 
How  many  trees  were  dead  ?     (b)  How  many  lived  ? 

2.  A  farmer  sold  2 1  bushels  of  oats ;  these  were  7  %  of 

all  he  raised ;  he  raised  bushels  and  had  bushels 

left. 

(c)  A  farmer  sold  98  bushels  of  oats;  these  were  7%  of 
all  he  raised.  How  many  bushels  did  he  raise  ?  (d)  How 
many  bushels  had  he  left  ? 

3.  A  grocer  sorted  200  bushels  of  apples;    12  bushels 

were  "  specked  "  and  the  remainder  were  sound ;  %  of 

the  apples  had  begun  to  decay  and  %   of  them  were 

sound. 

(e)  A  grocer  sorted  400  bushels  of  apples;  56  bushels 
were  "  specked  "  and  the  remainder  were  sound.  What  % 
of  the  apples  had  begun  to  decay  ?    (f)  What  %  were  sound  ? 

4.  Mr.  Briggs  earns  $900  a  year;  he  saves  8%  of  his 
wages ;  he  saves  dollars  and  spends  •  dollars. 

(g)  Mr.  Jones  earns  $1250  a  year;  he  saves  8%  of  his 
wages.     How  much  money  does  he  save  ? 

5.  Mr.  Piper  paid  $18  for  a  suit  of  clothes ;  this  was 
12%  of  his  month's  salary;  his  salary  was  dollars. 

(h)  Mr.  Brean  paid  $162  for  a  horse,  buggy,  and  harness; 
this  was  12%  of  his  year's  salary.  How  much  was  his 
yearly  salary  ? 

6.  George  earned  $300  ;  he  spent  $15  ;  he  spent  % 

of  what  he  earned. 

(i)  Henry  earned  $350 ;  he  spent  $63.  What  per  cent 
of  what  he  earned  did  he  spend  ? 


PART    1.  109 

REVIEW. 

A  manufacturer  employed  20  men.  The  daily  wages  of 
these  men  were  as  follows  :  Five  received  $3  00  each;  five 
received  $2.50  each;  five  received  $2.00  each;  and  five  re- 
ceived $1.50  each. 

(a)  Find  the  whole  amount  paid  to  these-  men  each  day. 

(b)  Find  the  average  daily  wages  of  the  men. 

(c)  Find  the  amount  paid  per  week  to  the  men  if  every 
man  worked  every  day  except  Sunday. 

(d)  Find  the  amount  paid  for  the  month  of  January,  if 
the  month  began  on  Monday  and  every  man  was  present 
every  day  except  the  Sundays  of  the  month. 

(e)  Divide  336  by  4f     (Change  to  ths.)     Story, 

(f)  Divide  336  by  4.8.     (336.0  ^  4.8  =.  ?)     Story. 
Compare  the  answers  to  problems  (e)  and  (f). 

(g)  Bought  a  ream  of  paper  for  $2.00  and  sold  it  at  18^ 
per  quire.     How  much  was  my  gain  ? 

(li)  Bought  a  box  (1  gross)  of  pens  for  75^  and  sold  them 
at  the  rate  of  4  pens  for  3  cents.     How  much  did  I  gain  ? 

(i)  If  10  bushels  of  potatoes  cost  $3.50,  how  much  will 
15  bu.  cost  at  the  same  rate  ? 

1.  Draw  a  diagram  of  a  piece  of  land  32  rods  by  25  rods, 

scale  ^  in.  to  a  rod.     The  diagram  is  inches  long  and 

and inches  wide. 

(j)  How  many  square  rods  in  the  piece  of  land  described 
in  problem  1  ?  (k)  How  many  acres  ?  (1)  The  perimeter 
of  the  field  is  how  many  rods  ?  (m)  How  much  will  it  cost 
to  build  a  fence  around  it  at  35^  a  rod  ?  (n)  How  much 
will  it  cost  to  plow  it  at  $1.25  an  acre  ? 

2.  7  is of  25.     7  twenty-fifths  are  —  hundredths. 


110  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    PROBLEMS. 

1.  A  woman  exchanged  3|-  lb.  of  butter  at  20^'  a  pound 
for  sugar  at  5^  a  pound.    She  received pounds  of  sugar. 

(a)  A  man  exchanged  9  cords  of  wood  @  $4.50  a  cord  for 
coal  at  $6.75  a  ton.  How  many  tons  of  coal  did  he  receive  ? 
(b)  How  many  pounds  of  coal  did  he  receive  ?  (c)  How 
many  loads  of  3000  lbs.  each  did  he  receive  ? 

2.  The  side  of  a  square  measures  5  ft.  8  in.;  the  peri- 
meter of  the  square  is  • feet  • inches. 

(d)  The  side  of  a  square  measures  47  ft.  10  in.  What  is 
the  measurement  of  the  perimeter  of  the  square  ? 

3.  An  estate  was  divided  among  a  widow  and  four  chil- 
dren ;  the  widow  received  1  third  of  the  estate ;  each  child 
received  1  fourth  of  what  remainded  after  the  widow  had 

received  her  share ;    each  child  received  — of  the 

estate. 

(e)  The  estate  mentioned  in  problem  3  was  worth  $7140. 
How  much  did  the  widow  receive  ?  (f)  How  much  did  all 
the  cliildren  together  receive  ?  (g)  What  each  child  received 
was  equal  to  what  part  of  what  the  widow  received  ? 

4.  The  width  of  an  oblong  is  8.  inches  ;  its  length  is  twice 

its  width ;    its   perimeter  is  inches ;  its   area  is  

square  inches. 

(h)  The  width  of  a  piece  of  land  in  the  shape  of  an  oblong 
is  32  rods ;  its  length  is  twice  its  width.  How  many  rods 
of  fence  required  to  enclose  it  ?  (i)  How  much  will  the 
fence  cost  at  35^  a  rod  ?  (j)  How  many  square  rods  in  the 
piece  ?  (k)  How  many  acres  in  the  piece  ?  (1)  What  is  it 
worth  at  $27.50  an  acre? 

5.  At  $1.40  a  yard,  5|-  yd.  of  carpet  cost  . 

(m)  At  $1.28  a  yard,  how  much  will  27^  yd.  of  carpet  cost  ? 


PART    I. 


Ill 


SIMPLE    NUMBERS  * 

1.  The  sum  of  all  the  odd  numbers  from  1  to  7  inclusive 
is  ;  of  all  the  even  numbers  from  2  to  8  inclusive, . 

(a)  Add  all  the  odd  numbers  from  1  to  37  inclusive. 

(b)  Add  all  the  even  numbers  from  2  to  38  inclusive. 

2.  5+4  +  6  +  7  +  3+8  +  6  +  5+4  +  3  +  5  +  6  +  3  + 
7-|.5-|-9_50=  . 

(c)  From  the  sum  of  9864,  792,  and  8756,  subtract  4598. 

(d)  Multiply  8754  by  27.  (e)  9328  x  46  = 
(f)  Multiply  5964  by  38. 
(h)  Multiply  3875  by  45. 
(j)  Multiply  874  by  126. 
(1)  Multiply  556  by  208. 
(n)  Multiply  734  by  160 

(p)  Divide  7618  by  52. 

(r)   Divide  9351  by  54. 

(t)   Divide  9636  by  55. 

(v)  Divide  7532  by  56. 

(x)  Divide  10336  by  57. 

(z)  Divide  3960  by  27. 
(bb)  Divide  6888  by  27. 
(dd)  Divide  6848  by  128. 
(ff)  Divide  5776  by  128. 
(hh)  Divide  2772  by  144. 
(jj)  Divide  4860  by  144. 
(11)  Divide  6440  by  160. 
(nn)  Divide  11800  by  160.  (oo)    5952  -^  160  = 

*TotheTeacheb.— If  pupils  are  inaccurate  in  their  work  in  simple  numbers, 
give  them  daily  practice  in  exercises  similar  to  those  api)earing  upon  this  page. 
Lead  the  pupils  to  feel  that  nothing  short  of  perfect  accuracy  is  highly  commend- 
able.   A  "  90  ji  pai)er ' '  iu  aiithmetic  is  a  failure.    A  "  90  jJ  accountant  "is  vjorthless. 


(g) 

3896  X  52  = 

(i) 

2543  X  69  = 

(k) 

963  X 

245  = 

(m)831x 

306  = 

(0) 

673  X 

250  = 

(q) 

12551 

-.-49  = 

(B) 

11292 

-.-48  = 

(u) 

9545^ 

-46  = 

(w) 

10635 

-.-45  = 

(y) 

10505 

-.-44  = 

(aa) 

8757 -f 

-27  = 

(cc) 

5649-. 

-27  = 

(ee) 

4768-. 

-128  = 

(gg) 

9696-. 

-128  = 

(ii) 

3960 -. 

-144  = 

(kk) 

7932  -. 

-144  = 

(mm) 

114280 

-.-160  = 

112  COMPLETE    ARITHMETIC. 

COMMON   FRACTIONS. 

1.  Add  i  |,  and  ^.     The  1.  c.  m.  of  9,  4,  and  6  is  — -. 

(a)  Add  346f,  375,  486|,  296,  and  855|. 

2.  From  12|  subtract  31^.     15|  -  2^  = 

(b)  From  2381  subtract  154i-.     (c)  596|  -  148|-  = 

3.  Multiply  3|  by  4.  4.  Multiply  7f  by  8. 
(d)  Multiply  537|  by  7.       (e)  Multiply  375f  by  9. 

5.  Multiply  12  by  2f  6.  Multiply  11  by  3^. 

(f)  Multiply  874  by  2i.       (g)  Multiply  954  by  3J. 

7.  Multiply  11^  by  4i.     This  means  ■ . 

(h)  Multiply  3731-  ^y  4i.    (i)  Multiply  525|-  by  4f 

•8.  Divide  8^  by  |.      (Change  8^  to  ths.) 

(j)  Divide  48i  by  |.  (k)  Divide  17|  by  |. 

9.  Divide  6f  by  If  6 f  =  ^.     1^  =^. 

(1)  Divide  75f  by  If  (m)  Divide  82f  by  If 
10.  Divide  1-J  (-J)  by  3.  11.  Divide  19  J  by  3.* 
(n)  Divide  577J  by  3.         (o)   Divide  625-J  by  3. 

12.  James  rode  his  wheel  at  the  rate  of  9|  mi.  an  hour: 

In  4  hours  he  rode  and miles.     In  ^  hour 

he  rode  and miles.     In  4^  hours  he  rode 

and  miles. 

A  locomotive  moved  at  the  rate  of  49|  miles  an  hour : 
(p)  How  far  did  it  move  in  4  hours  ?    (q)  In  5  hours  ? 
(r)    How  far  did  it  move  in  ^  of  an  hour?  (s)  In  ^  of  an  hr.? 
(t)    How  far  did  it  move  in  4^  hours  ?     (u)  In  5}  hours  ? 

13.  If  I  of  a  yard  of  print  is  sufficient  for  a  child's  apron, 
8 J  yards  are  sufficient  for  aprons.     8^  ^  |  = 

(v)  79|-  yards  divided  by  2  J  yards. 

*  Do  not  change  19J  to  sixths.    Say,  rather,  1  third  of  191  is  6,  with  a  remainder  of 
11;  1  third  of  IJ  (5)  is  i\. 


PART   1, 


113 


DECIMAL   FKACTIONS/ 


2  _ 
■3  - 

1  _ 
T- 

i  = 
i  = 

2  _ 

f- 

4 


tenths.  ^  = 
tenths.  -1-  = 
tenths.    |-  =  —  hundredths.    -I  = 


hundredths.    ^ 
hundredths.    -^ 


■  tenths.  ^  = 

•  tenths.  |-  = 

•  tenths.  ^  = 

•  tenths.  I  = 
tenths.  |-  = 


6 

hundredths.  ^ 

hundredths,  f 

hundredths.  ^ 

hundredths,  f 

hundredths.  f 

^  =  —  tenths,    f  =  —  hundredths.  |- 

-J  =  —  tenths.    -J  =  —  hundredths.  -J 

hundredths.  | 

hundredths.  4" 

hundredths.  ^ 

hundredths,  -i- 


3  _ 


6 

■6  ~ 
1  _ 
T  — 

i= 

1  — 
-w  — 


5  _ 


•  tenths. 

■  tenths.  ^  = 

•  tenths.  ^  =. 

■  tenths.  ^  = 


—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 

—  thousandths. 


(1)  Tell  the  meaning  of  each  of  the  following,  (2)  solve,  and  (3) 
tell  a  suggested  number  story. 

(a)  Multiply  $345  by  .7. 

(c)  Multiply  $345  by  .08. 

(e)  Multiply  $345  by  3.28. 

(g)  Multiply  $345  by  .025. 

(i)  Divide  $648  by  $9. 

(k)  Divide  $648  by  $.9. 

(m)  Find  the  cost  of  7.2  tons  of  coal  at  $4.75  a  ton. 

(n)  I  paid  $38.25  for  coal  at  $4.50  per  ton.  How  many 
tons  did  I  buy  ? 

(0)  I  paid  $663  for  156  tons  of  coal.  What  was  the  price 
per  ton  ? 

*  The  pupil  will  probably  need  a  pencil  in  the  solution  of  some  of  the  problems 
in  the  first  fourteen  lines.  After  he  is  able  to  fill  the  blanks,  require  him  to  write 
them  as  suggested  in  the  following .  i  =  .5;  J  =  .50;  i  =  .500;  J  =  .3J;  i  =  33i; 
i  =  .338J,  etc. 


(b)  Multiply  $345  by  5.7. 
(d)  Multiply  $345  by  .28. 
(f)  Multiply  $345  by  .005. 
(h)  Multiply  $345  by  .325. 
(j)  Divide  $648  by  9. 
(1)  Divide  $648  by  $.09. 


114  COMPLETE    ARITHMETIC. 

DENOMINATE    NUMBERS. 

100  lb.  =  1  hundredweight  (cwt.). 

1.  At  60<f  per  cwt.,  1  ton  of  bran  costs  dollars. 

2.  At  40^  per  cwt.,  |-  ton  of  feed  costs  dollars. 

3.  At  60^  per  cwt.,  ^  ton  of  corn  meal  costs  . 

(a)  At  $2.50  per  cwt.,  how  much  will  f  of  a  ton  of  flour 
cost  ?     (b)  ^V  of  a  ton  ?     (c)  |  of  a  ton  ? 

4.  At  $2.40  per  cwt.,  50  lb.  of  flour  are  worth  . 


5.  At  50^  per  cwt,  1050  lb.  cost ;  850  lb.  cost ; 

1250  lb.  cost ;  950  lb.  cost ;  1150  lb.  cost . 

(d)  At  $2.70  per  cwt,  how  much  will  1450  lb.  cost  ?     (e) 
1950  lb.  ?     (f)  1350  lb.  ? 

6.  At  $2.40  per  cwt,  25  lb.  of  flour  are  worth  . 


7.  At  60^  per  cwt,  25  lb.  are  worth  cents ;  75  lb. 

are  worth  cents;  175  lb.  are  worth  . 

(g)  At  $2.60  per  cwt,  how  much  wiU  1225  lb.  cost  ?    (h) 
1275  lb.?     (i)  1525  lb.? 

8.  At  $2.50  per  cwt,  20  lb.  of  flour  are  worth cents. 

9.  At  60^  per  cwt,  20  lb.  are  worth  cents;  40  lb. 

are  worth  cents ;  60  lb.  are  worth  cents ;  80  lb. 

are  worth  cents. 

(j)  At  $2.50  per  cwt,  how  much  will  1220  lb.  cost;  (k) 
1240  lb.?     (1)  1260  lb.? 

10.  At  $2.30  per  cwt,  10  lb.  of  flour  are  worth  . 

11.  At  80^  per  cwt,  10  lb.  of  corn  are  worth  cents; 

20  lb.  are  worth  cents ;  30  lb.  are  worth  cents ;  40 

lb.  are  worth cents ;  50  lb.  are  worth cents ;  60  lb. 

are  worth cents ;  70  lb.  are  worth cents. 

(m)  At  $2.50  per  cwt.,  how  much  will  1210  lb.  cost? 


PAET    I.  115 


MEASUEEMENTS. 


1.  Draw  carefully  upon  your  slate  or  paper,  on  a  scale  of 
j-  in.  to  the  foot,  a  diagram  of  the  floor  of  a  room  that  is  17 
ft.  long  and  14  ft.  wide. 

(a)  How  many  square  feet  in  the  floor  ? 

(b)  How  many  feet  of  flooring  will  be  required  for  the 
floor  if,  on  account  of  the  waste  and  the  loss  in  matching, 
you  must  buy  1  fourth  more  feet  of  flooring  than  there  are 
square  feet  to  be  covered  ? 

(c)  Making  no  allowance  for  doors,  how  many  feet  in 
length  are  the  mop-boards  of  the  room  ? 

(d)  If  this  floor  is  to  be  covered  with  a  carpet  that  is  1 
yard  wide  and  the  strips  are  to  "run  lengthwise"  of  the 
room,  how  many  feet  long  will  each  strip  be  ? 

(e)  How  many  strips  must  be  purchased  ? 

(f )  How  much  of  one  strip  must  be  turned  under  or  cut 
off?  " 

(g)  How  many  feet  (in  length)  of  carpet  must  be  pur- 
chased if  there  is  no  waste  in  matching  the  strips  ? 

(h)  How  many  yards  must  be  purchased  ? 

(i)  How  much  will  the  carpet  cost  at  60^  a  yard  ? 

(j)  If  the  figure  of  the  carpet  is  such  that  there  is  waste 
in  matching,  more  carpet  must  be  purchased  than  would 
otherwise  be  required.  How  much  more  if  the  waste  on 
each  strip  (except  the  first)  is  1  foot  ? 

(k)  How  much,  including  the  waste,  will  the  carpet  cost 
at  60^  a  yard  ? 

(1)  If  the  same  floor  is  to  be  covered  with  carpeting  that 
is  1  yard  wide,  costing  40^  a  yard,  the  strips  to  run  length- 
wise, and  the  waste  in  matching  being  6  in.  on  each  strip 
except  the  first,  how  much  will  the  carpet  cost  ? 


116  COMPLETE    AKITHMETIC. 

RATIO   AND    PROPORTION. 

1.  Six  inches  are of  a  rod. 

2.  Twelve  inches  are of  a  rod. 

3.  One  foot  and  6  inches  are of  a  rod. 

4.  Two  feet  are of  a  rod. 

6.  Two  feet  and  six  inches  are of  a  rod. 

6.  Three  feet  and  six  inches  are of  a  rod. 

7.  Four  feet  and  six  inches  are of  a  rod. 

8.  One  rod  is of  a  mile.  2  rods  are  . 

9.  Three  rods  are of  a  mile.     4  rods  are  . 

(a)  Twelve  rods  are  what  part  of  a  mile  ?     (b)  10  rods  ? 

(c)  20  rods  ?     (d)  30  ?     (e)  40  rods  ?  .  (f)  60  rods  ? 

10.  200  rods  are of   a   mile.     1  mile  is  

of  200  rods,  or  and times  200  rods. 

11.  12  feet  are of  a  rod.     1  rod  is 

of  12  feet,  or  and times  12  feet. 

12.  Two  thirds  of  7^  feet  (-i/)  are  1  half  of  feet. 

13.  Two  thirds  of  4J  ft.  (|)  are  1  half  of  feet. 

(g)   Two  thirds  of  25|-  ft.  are  1  half  of  how  many  feet  ? 
(h)  Two  thirds  of  37^  rd.  are  1  half  of  how  many  rods  ? 

14.  If  a  piece  of  rubber  hose  5J  feet  long  is  worth  60^,  a 
piece  16^  feet  is  worth  . 

(i)    If  it  is  worth  $1.65  to  make  5^  ft.  of  concrete  walk, 
how  much  is  it  worth  to  make  IQ^  ft.  of  similar  walk  ? 

15.  Twenty-five  rods  are of  37^  rods.      37|- 

rods  are of  25  rods,  or  and times 

25  rods. 

(j)    If  it  costs  $75.30  to  make  25  rods  of  road,  how  much 
will  it  cost  to  make  37^  rods  at  the  same  rate  ? 


PART    I.  117 

PERCENTAGE. 

{!) 

1.  50%  more  than  60  is *        50%  less  than  60  =t 

2.  25  %  more  than  60  is .  25  %  less  than  60  = 

3.  20%  more  than  60  is .  20%  less  than  60  = 

(a)  50%  more  than  $846  =  (b)  50%  less  than  $846  = 

(c)  25%  more  than  $576  =  (d)  25%  less  than  $576  = 

(e)  2 0  %  more  than  $475  =  (f)  2 0  %  less  than  $475  = 

{2) 

4.  60  is  50%  more  than 4  60  is  50%  less  than .§ 

5.  60  is  2 5  %  more  than .     60  is  2 5  %  less  than . 

6.  60  is  20%  more  than .     60  is  20%  less  than . 

(g)  $420  is  50%  more  than  how  many  dollars  ? 
(h)$420  is  50%  less  than  how  many  dollars  ? 

(i)  $420  is  25%  more  than  how  many  dollars  ? 
(j)  $420  is  25%  less  than  how  many  dollars  ? 
(k)  $42  0  is  2  0  %  more  than  how  many  dollars  ? 
(1)  $420  is  20%  less  than  how  many  dollars  ? 

(5) 

7.  50  is %  more  than  40. ||  30  is %  less  than  40.^ 

8.  60  is %  more  than  50.     35  is %  less  than  70. 

9.  30  is  — -%  more  than  20.    40  is %  less  than  50. 

(m)$660  is  how  many  per  cent  more  than  $440  ? 

(n)  $630  is  how  many  per  cent  less  than  $840  ? 

*  50^  of  60  more  than  60.    50^  =  J.    J  of  60  =  30.    60  +  30  =  90. 

t  50/»  of  60  less  than  60.     50^  =  J.    ^  of  60  =  30.    60  -  30  =  30. 

:j:  Think  of  x  as  standing  for  the  number  sought  {the  base).  Then  60  =  x  and  1 
half  of  X,  or  f  of  x.  Since  60  =  i  of  x,  1  half  of  x  equals  20,  and  the  whole  of  a;  =  2 
times  20,  or  40. 

g  60  =  re  less  1  half  of  x,  or  i  of  x, 

II  50  is  10  more  than  40,  or  Jg  o/AO  more  than  hO.    \%  =  \  —  'i^ff,. 

II  30  is  10  less  than  40,  or  l%  of  UO  less  than  W.    i8  =  i  =  255«. 


118  COMPLETE    ARITHMETIC. 


PERCENTAGE. 


1.  Ira  earns  80  cents  a  day;  Ernest  earns  25%  more 

than  Ira;  Ernest  earns  a  day.     Arthur  earns  25%  less 

than  Ira ;  Arthur  earns  a  day. 

(a)  Mr.  James  earns  $860  a  year;  Mr.  Brown  earns  25% 
more  than  Mr.  James.  How  much  does  Mr.  Brown  earn  ? 
(b)  Mr.  White  earns  2  5  %  less  than  Mr.  James-.  How  much 
does  Mr.  White  earn  ? 

2.  Peter  earns  $60  a  month  ;  this  is  25%  more  than  Paul 
earns ;  Paul  earns  a  month.* 

(c)  Mr.  Harris  earns  $1200  a  year ;  this  is  25%  more  than 
Mr.  Williams  earns.     How  much  does  Mr.  Williams  earn  ? 

3.  Eichard  earns  $60  a  month;  this  is  25%  less  than 
Harry  earns ;  Harry  earns  a  month,  f 

(d)  Mr.  Smith  earns  $840  a  year;  this  is  25%  less  than 
Mr.  Eice  earns.     How  much  does  Mr.  Eice  earn  ? 

4.  George  earns  $40  a  month  ;  Joseph  earns  $50  a  month ; 

George  earns  per  cent  less  than  Joseph ;  Joseph  earns 

per  cent  more  than  George.  J 

(e)  Mr.  Dow  earns  $1000  a  year;  Mr.  Wheeler  earns 
$1250  a  year.  Mr.  Wheeler  earns  how  many  per  cent  more 
than  Mr.  Dow  ?  (f)  Mr.  Dow  earns  how  many  per  cent  less 
than  Mr.  Wheeler  ? 

15  is  %  more  than  10.         10  is  %  less  than  15. 

20  is  %  more  than  15.         15  is  %  less  than  20. 

25  is  %  more  than  20.        20  is  %  less  than  25. 

*  Think  of  x  as  standing  for  what  Paul  earns.  Then  $60  =  a;  and  1  fourth  of  x,  or 
5  fourths  of  X.    If  $60  is  J  of  x,  then  x  is dollars. 

t  Let  X  stand  for  what  Harry  earns.  Then  $60  =  a;  less  1  fourth  of  x,  or  i  of  a;.  If 
860  is  i  of  X,  then  x  is dollars. 

X  In  the  first  part  of  this  problem  Joseph's  money  is  the  base,  and  in  the  second 
paxt  George's  money  is  the  base.  Joseph  earns  $10  more  than  $40,  or  I  of  $40  more 
than  $40.    George  earns  $10  less  than  $50,  or  J  of  $50  less  than  $50. 


PART    I.  119 


REVIEW. 


(a)  Find  the  sum  of  all  the  odd  numbers  from  59  to  73 
inclusive. 

(b)  Find  the  sum  of  all  the  even  numbers  from  64  to  86 
inclusive. 

(c)  Divide  12  by  f.         (d)  Multiply  12  by  |. 
Compare  the  answers  to  problems  (c)  and  (d). 

1.  -|^  =  hundredths.         ^  =  thousandths. 

(e)  Change  |  to  hundredths.  (f)  Change  f  to  lOOOths. 
(g)  Change  f  to  hundredths.  (h)  Change  |  to  lOOOths. 
(i)  Change  -J  to  hundredths.         (j)  Change  |-  to  lOOOths. 

2.  At  $3  per  cwt.,  500   lb.  of  pork  cost  dollars; 

650  lb.  cost  ;  510  lb.  cost  ;  520  lb.  cost  . 

(k)  At  S3.60  per  cwt.,  what  is  the  value  of  500  lb.  of 
pork?     (1)  Of  550  lb.?     (m)  Of  510  lb.?     (n)  Of  520  lb.? 

3.  At  60^  a  gross,  1  dozen  pens  cost cents ;  3  dozen 

cost  cents;  7  dozen  cost  cents. 

(o)  At  $5.28  a  gross,  what  is  the  value  of  1  dozen  fruit- 
cans  ?     (p)  Of  8  dozen  ? 

(q)At  $7.20  a  gross,  what  is  the  cost  of  3  dozen  fruit- 
cans  ?     (r)  Of  3  cans  ? 

4.  A  floor  15  feet  by  17  feet  is  to  be  covered  with  carpet 
that  is  1  yard  wide.     If  the  strips  are  laid  lengthwise  of  the 

room,  and  there  is  no  waste  in  matching  the  figure,  

strips  will  be  required,  each  of  which  is  feet  long. 

(s)  How  many  yards  of  carpet  will  be  required  for  the 
room  described  in  problem  4  ?  (t)  How  much  will  it  cost  at 
90^  a  yard? 


120  COMPLETE    ARITHMETIC. 

MISCELLANEOUS   PROBLEMS. 

1.  The  state  of  Illinois  is  about  380  miles  long  and  225 
miles  wide  at  its  widest  part.     A  map  of  Illinois  drawn 

upon  a  scale  of  10  miles  to  the  inch  must  be  inches 

long  and  and inches  wide. 

2.  A  certain  county  of  Illinois  appears  upon  the  map 
described  in  problem  1  as  a  rectangular  figure  1.8  inches  by  3 

inches.     The    county   is  miles    wide    and  miles 

long. 

(a)  How  many  square  miles  in  the  county  described  in 
problem  2  ? 

Tell  the  amount  of  change  that  should  be  given  in  each  of  the 
following  instances : 

3.  Bought  4|-  lb.  meat  @  12^ ;  gave  the  salesman  $1. 

4.  Bought  3^  lb.  cheese  @  16^-;  gave  the  salesman  $1. 

5.  Bought  2|-  lb.  cheese  @  16^;  gave  the  salesman  50^. 

6.  Bought  2|-  dozen  eggs  @  14^ ;  gave  the  salesman  $2. 

(b)  Bo't  2.3  tons  coal  @  $6.50 ;  gave  the  salesman  S20. 

(c)  Bo't  1.4  tons  hay  @  $9.50 ;  gave  the  salesman  $15. 

(d)  Bo't  3^  cords  wood  @  $4.50  ;  gave  the  salesman  $16. 

7.  Byron  bought  8  melons  for  50^;  he  sold  one  half  of 
them  at  10^  each  and  the  other  half  at  5^  each;  he  gained 
cents. 

(e)  A  drover  bought  36  sheep  for  $125  ;  he  sold  one  half 
of  them  at  $4  each  and  the  other  half  at  $4.50  each.  How 
much  did  he  gain  ? 

8.  A  house  rents  for  $10  per  month  ;  the  rent  for  one  year 
and  six  months  is  dollars. 

(f)  A  house  rents  for  $32  per  month.  How  much  is  the 
rent  for  2  years  and  4  months  ? 


PAKT   I. 

SIMPLE   NUMBERS  * 

TICKETS   OF   ADMISSION   SOLD   AT   A   COUNTY  FAIR. 


121 


Price 

Wed. 

Thur. 

Fri. 

Sat. 

Total. 

Children's  Tickets 

Adults'  Tickets 

15^ 

25?* 
50^ 

1645 

2243 

143 

123 

1154 

1754 

174 

175 

3561 

3871 

186 

162 

1424 

2124 

75 

137 

(e) 

Single  Carriage  Tickets. . 
Double  Carriage  Tickets . 

(a) 

(b) 

(c) 

(d) 

(a)  to  (d)  Find  the  number  of  tickets  sold  each  day. 

(e)  Find  the  number  of  children's  tickets  sold. 

(f)  Find  the  number  of  adults'  tickets  sold. 

(g)  Find  the  number  of  single  carriage  tickets  sold, 
(h)  Find  the  number  of  double  carriage  tickets  sold. 

(i)  Find  the  sum  of  (a)  to  (d)  inclusive,  and  (e)  to  (h)  in- 
clusive. 

(j)  Find  the  receipts  for  tickets  sold  Wednesday;  (k) 
Thursday;  (1)  Friday;  (m)  Saturday;  (n)  total  receipts  for 
tickets  sold. 

(o)  Find  the  receipts  for  all  the  children's  ticket  sold  dur- 
ing the  week  ;  (p)  adults'  tickets ;  (q)  single  carriage  tickets ; 
(r)  double  carriage  tickets. 

(s)  Find  the  sum  of  (o),  (p),  (q),  and  (r),  and  compare 
this  sum  with  the  answer  to  problem  (n). 


-  days.f 

-  days4 

-  days, 
(t)  How  many  days  from  June  28th  to  Dec.  25th  ? 

From  April  20  th  to  Dec.  10  th  ? 


1.  From  June  28th  to  July  3rd  it  is 

2.  From  June  28th  to  Aug.  3rd  it  is 

3.  From  June  28th  to  Sept.  3rd  it  is 


(ui 


*  Insist  upon  accuracy :  "  90  per  cent  of  accuracy ' 
t  To  June  30th  is  2  days ;  to  July  3rd,  3  more  days. 
t  Two  and  81  and  3. 


is  failure. 


122  COMPLETE    ARITHMETIC. 


COMMON   FRACTIONS. 
Reduce  the  following  fractions  to  their  lowest  terms : 

124—  18—  21—  24_  15—  18    — 

^'     3^   —  2 T  —  J-Q   —  T¥  —  3" 5"  —  TS  - 

(a)  111=     .(b)Hf=      (c)AV=       (d)/A  = 

Reduce  the  following  improper  fractions  to  whole  or  mixed 
numbers : 


2.  V-  = 

=      -V-  = 

l*  = 

u  = 

(e)^  = 

(0 

l^A  = 

(g)  H^  = 

=        (h) 

H^  = 

Reduce  the  following  mixed  numbers  to 

improper  : 

fractions : 

3.  7f  = 

9f  = 

8|  = 

12i  = 

iif  = 

101  = 

(i)  564  = 

(J) 

74f  = 

(k)  86| 

=     (1) 

73f  = 

Reduce  the  following  fractions  to  equivalent  fractions  having  a 
common  denominator : 

4.  -|,  ^,  and  ^.     The  1.  c.  m.  of  8,  4,  and  6  is  . 


(m)  I-  and  -J.         (n)  ^V  and  f .         (o)  ^%  and  3%. 


(p)  Change  47  to  a  fraction  whose  denominator  is  6. 

6.  The  sum  of  two  numbers  is  78 ;  one  of  the  numbers 
is  32 ;  the  other  number  is  . 

(q)  The  sum  of  two  fractions  is  f ;  one  of  the  fractions  is 
|.     What  is  the  other  fraction  ? 

7.  If  it  takes  f  of  a  yard  of  cloth  to  make  one  vest,  from 
three  yards  vests  can  be  made. 

(r)  How  many  vests,  each  containing  f  of  a  yard,  can  be 
made  from  36  yards?  (s)  From  48  yards?  (t)  From  GO 
yards  ? 


PART   I.  123 


DECIMAL   FRACTIONS. 


Change  the  following  common  fractions  to  decimal  fractions : 

Il_  1—  3—  2—  3—  4_ 

^'    ^  —  T-  T-  T-  6^-  J  - 

(a)i  =  *  (b)|=  (c)-J=  (d)|=  (e)^  = 
(f)  f  =  (g)  f  =  (h)  f  =  (i)  4  =  (J)  f  = 
(k)i=        (1)1=        (mH=        (n)|=        (0)1  = 

Change  the  following  decimal  fractions  to  common  fractions, 
and  reduce  to  their  lowest  terms : 

2.  .8  =  .5  =  A=  .25  r=  .75  =  .121-  ^  |  .3^  ^ 
(p)  .125  =  (q)  .375  =  (r)  .625  =  (s)  .875  = 
(t).175=         (u).225=         (v).325=         (w).675  . 

(1)  Tell  the  meaning  of  each  of  the  following,  (2)  solve,  and  (3) 
tell  a  suggested  number  story :  J 

(x)  Multiply  $475  by  .9.  (y)  Multiply  $475  by  2.9. 

(z)  Multiply  $534  by  .07.  (aa)  Multiply  $534  by  .37. 

(bb)  Multiply  $534  by  2.37.  (cc)  Multiply  $534  by  .003. 

(dd)  Multiply  $534  by  .043.  (ee)  Multiply  $534  by  .243. 

(ff)    Divide  $724  by  $8.  (gg)  Divide  $724  by  8. 

(hh)  Divide  $724  by  $.8.  (ii)  Divide  $724  by  $.08. 

(jj)    At  $12.50  per  ton,  find  the  cost  of  3.7  tons  of  hay. 
(kk)  At  $9.75  per  ton,  find  the  cost  of  3.4  tons  of  hay. 

*  Change  these  to  thousandths.  There  are  1(XX)  thousandths  in  a  whole  (a  unit) ; 
in  J  of  a  unit  there  are  J  of  1000  thousandths.  This  answer  may  be  written,  .333J  or 
333+. 

t  Observe  that  you  can  divide  both  the  numerator  and  denominator  of  12J  hun- 
dredths by  12i. 

%  Do  not  omit  the  number  story,  or  business  problem.  The  number  story  for 
problem  (x)  might  be :  Mr.  Hoyt  bought  .9  of  an  acre  of  land  at  $lt75  an  acre;  the  land 
cost  him dollars. 


124  COMPLETE    ARITHMETIC. 

DENOMINATE   NUMBEKS. 
Find  the  cost :  * 

1.  3000  lb.  hay  at  $6  per  ton.  2.  500  lb.  hay  @  S6. 

3.  1500  lath  at  $3  per  M.  4.  250  lath  @  S3. 

5.  1250  lb.  pork  at  $4  per  cwt.  6.  125  lb.  pork  @  $4. 

7.  2500  lb.  coal  at  $5  per  ton.  8.  400  lb.  coal  @  $5. 

9.  650  lb.  beef  @  $5  per  cwt.  10.  25  lb.  beef  at  $5. 

11.  2500  brick  @  $8  per  M.  12.  250  brick  @  $8. 

Find  the  value : 

13.  1  brick  at  $8  per  M.  14.  100  brick  @  $8. 

15.  1  lb.  hay  at  $10  per  ton.  16.  100  lb.  hay  @  $10. 

17.  1  lb.  of  beef  @  $4  per  cwt.  18.  10  lb.  beef  @  $4. 

(a)  2150  brick  at  $8  per  M.  (b)  625  brick  @  $8. 

(c)   2150  lb.  pork  at  $4  per  cwt.  (d)  625  lb.  pork  @  $4. 

(e)   2150  lb.  hay  at  $10  per  ton.  (f)  650  lb.  hay  @  $10. 

(g)   3240  lath  at  $3  per  M.  (h)  750  lath  @  $3. 

(i)    3240  lb.  coal  @  $5.50  per  ton.  (j)  760  lb.  coal  @  $5.50. 

(k)  3240  lb.  beef  @  $6.50  per  cwt.  (1)  86  lb.  beef  @  $6.50. 

19.  At  60^  a  gross,  2  doz.  buttons  cost  cents. 

(m)  At  $1.60  per  gross,  75  doz.  buttons  cost  how  much  ? 

(n)  Make  a  receipted  bill  of  the  following  goods  sold  by 
yourself  to  Christopher  Columbus : 

Sept.  1,  1230  lb.  pork  @  $4.60  per  cwt,  and  1080  lb. 
beef  @  $6.75  per  cwt. 

20.  At  $2.40  per  ream,  1  quire  of  paper  costs  cents ; 

12  sheets  cost  cents;  6  sheets  cost  cents. 

(o)  Bought  paper  @  $2.40  per  ream  and  sold  it  at  15^  a 
quire.     What  was  the  gain  on  3  reams  ? 

(t))  At  7^  a  quire,  how  much  will  1440  sheets  of  paper  cost  ? 

♦  Pork  and  beef  are  ustially  bought  and  sold  by  the  hundredweight  (cwt.),  hay 
and  coal  by  the  ton,  and  lath  and  brick  by  the  thousand  (M.). 


PART   I.  125 

MEASUREMENTS. 

1.  Draw  carefully  upon  your  slate  or  paper,  on  a  scale  of 
^  in.  to  the  foot,  a  diagram  of  a  rectangular  grass  plot  50  ft. 
by  40  ft.  Kepresent  a  gravel  walk  5  feet  wide  just  outside 
the  grass  plot  and  extending  entirely  around  it.  The  dia- 
gram of  the  grass  plot  is  and inches  long 

and  inches  wide.     The  width  of  the  walk  as  shown  in 

the   diagram  is of  an  inch.     The  entire  length 

of  the  figure  including  the  diagram  of  the  grass  plot  and 
walk  is  and  inches. 

(a)  How  many  square  feet  in  the  grass  plot  described  in 
problem  1  ?     (b)  How  many  square  yards  ? 

(c)  The  perimeter  of  the  grass  plot  described  in  problem  1 
is  how  many  feet  ?     (d)  How  many  yards  ? 

(e)  The  perimeter  of  the  outside  of  the  gravel  walk  de- 
scribed in  problem  1,  is  how  many  feet  ?  (f)  How  many 
yards  ? 

(g)  The  outside  perimeter  of  the  gravel  walk  described  in 
problem  1  is  how  much  greater  than  the  inside  perimeter  ? 
(h)  How  many  square  feet  in  the  walk  ?  (i)  How  many 
square  yards  in  the  walk  ?  (j)  How  much  did  it  cost  to 
make  the  walk  at  54^  per  square  yard  ? 

2.  A  floor  14  ft.  by  17  ft.  is  to  be  covered  with  a  carpet 

that  is  one  yard  wide.     If  the  breadths  are  1 7  feet  long, 

breadths  will  be  required.     If  the  breadths  are  14  ft.  long, 
breadths  will  be  required. 

(k)  If  there  is  no  waste  in  matching,  how  many  yards  of 
carpet  will  be  required  for  the  room  described  in  problem  2 
if  17-foot   strips  are  used?     (1)  How  many  yards  if  14-foot 
strips  are  used  ? 


126  COMPLETE    ARITHMETIC. 

RATIO   AND    PROPORTION. 

1.  One  cwt.  is of  a  ton.         2  cwt.  = 

2.  Three  cwt.  are of  a  ton.         4  cwt.  = 

3.  Five  cwt.  are of  a  ton.         6  cwt.  = 

4.  Seven  cwt.  are of  a  ton.         8  cwt.  = 

5.  Fifteen  cwt.  are of  a  ton.     One  ton  is 

of  15  cwt.,  or  and times  15  cwt. 

6.  Sixteen  cwt.  are of  a  ton.     One  ton  is  

of  16  cwt.,  or  and times  16  cwt. 

7.  Eight  cwt.,  are  of  a  ton.     One  ton  is  

of  8  cwt.,  or  and  times  8  cwt. 

8.  Twelve  cwt.  of  hay  is of  a  ton  of  hay. 

(a)  If  one  ton  of  hay  is  worth  $12.75,  how  much  is  12 
cwt.  of  hay  worth  at  the  same  rate  ? 

9.  Eighteen  cwt.  of  coal  is of  a  ton  of  coal. 

(b)  If  one  ton  of  coal  is  worth  $6.50,  how  much  is  18 
cwt.  of  coal  worth  at  the  same  rate  ? 

10.  Six  cwt.  of  flour  is of  a  ton  of  flour.     One 

ton  of  flour  is of  6  cwt.  of  flour,  or  and  

times  cwt. 

(c)  If  6  cwt.  of  flour  is  worth  $12.30,  how  much  is  1  ton 
of  flour  worth  at  the  same  rate  ? 

11.  Forty  pounds  are  of  1  cwt. 

(d)  If  1  cwt.  of  sugar  is  worth  $5.25,  how  much  is  40  lb. 
of  sugar  worth  ? 

12.  Eighty  pounds  are  of  1  cwt.      1  cwt.  is 

of  80  lb.,  or times  80  lb. 

(e)  If  80  lb.  of  beef  is  worth  $5.60,  how  much  is  1  cwt. 
of  beef  worth  at  the  same  rate  ? 


PART  I.  127 

PERCENTAGE. 

1.  40%  more  than  25  is *    40%  less  than  25  is . 

2.  16f  %  more  than  30  is  .     16f  %  less  than  30  is 

3.  33i%  more  than  36  is  .     33i%  less  than  36  is 

(a)  40%  more  than  $570  =    (b)  40%  less  than  $570  = 
(c)  16|%  more  than  $834  =  (d)  16f  %  less  than  $834  = 
(e)  33-1-%  more  than  $726  =  (f)  33^%  less  than  $726  = 

(2) 

4.  49  is  40%   more  than  .f     36  is  40%  less  than 

— 4 

5.  42  is  16|%  more  than  .     40  is  16f  %  less  than 

6.  48  is  33i-%  more  than  .     30  is  33^%  less  than 

(g)  $245  is  40  per  cent  more  than  how  many  dollars  ? 
(h)  $831  is  40  per  cent  less  than  how  many  dollars  ? 
(i)  $658  is  16|  per  cent  more  than  how  many  dollars  ? 
(j)  $645  is  16f  per  cent  less  than  how  many  dollars  ? 

(3) 

7.  70  is %  more  than  50.§  50  is  — %  less  than  60  || 

8.  40  is %  more  than  30.    30  is %  less  than  50. 

(k)  $450  is  how  many  per  cent  less  than  $750  ? 

(1)  $400  is  how  many  per  cent  less  than  $480  ? 
(m)$480  is  how  many  per  cent  more  than  $360  ? 

*40^of\vhat?    iO%=l.    §  of  25  =10.    25  +  10  =  35. 

t Think  of  x  as  standing  for  the  number  sought  (the  base).  Then  49  =  z  and  2 
fifths  of  z,  or  I  of  X.  Since  49  is  g  of  a;,  1  fifth  of  x  is  7,  and  the  whole  of  a;  =  5  times  7, 
or  85. 

$  36  =  a;  less  2  fifths  of  x,  or  f  of  x. 

§70  is  20  more  than  50,  or  §g  of  50  more  than  50.    ig  '=  §.    i  =  40$^. 

i  50  is  10  less  than  60,  or  Jg  of  60  less  than  60.    |g  =  J,  or  161  %. 


128  COMPLETE   ARITHMETIC. 

PEKCENTAGE. 

1.  Hiram  had  $50 ;  Samuel  had  $40 ;  Hiram  had % 

more  than  Samuel;  Samuel  had  %  less  than  Hiram. 

(a)  Mr.  Cooper  has  880  bushels  of  corn;  Mr.  Judd  has 
660  bu.  Mr.  Judd  has  how  many  per  cent  less  than  Mr. 
Cooper  ?  (b)  Mr.  Cooper  has  how  many  per  cent  more  than 
Mr.  Judd? 

2.  In  1896  a  farmer  raised  60  bu.  of  oats  to  the  acre; 
this  was  a  50%  better  yield  than  he  had  in  1895;  in  1895 
he  raised  bushels  to  the  acre.* 

(c)  In  1896  a  farmer  sold  726  bushels  of  wheat;  this  was 
50%  more  than  he  sold  in  1895.  How  many  bushels  did 
he  sell  in  1895? 

3.  William  earns  $30  a  month;    Benjamin  earns  40% 

more  than  Wilham ;  Benjamin  earns  dollars  a  month. 

Joseph  earns  50%  more  than  Benjamin;  Joseph  earns  

dollars  a  month. 

(d)  Isaac  earns  $27.50  a  month;  Ralph  earns  40%  more 
than  Isaac.  How  much  does  Ralph  earn  per  month  ?  (e) 
Clinton  earns  50%  more  than  Ralph.  How  much  does 
CHnton  earn  per  month  ? 

4.  Good  cheese  is  worth   18^  a  pound;   this  is   50% 

higher  than  it  was  1  year  ago ;  a  year  ago  it  was  worth 

cents. 

(f)  A  certain  brand  of  flour  is  worth  $5.55  per  barrel. 
If  this  is  50%  more  than  it  was  worth  a  year  ago,  what  was 
it  then  worth  ? 

18  is  %  more  than  12.      12  is  %  less  than  18. 

24  is  %  more  than  18.       18  is  %  less  than  24. 

*  Think  of  x  as  standing  for  the  yield  of  1895.  Then  60  bu.  =  a;  and  1  half  of  x, 
or  I  of  X.    Since  60  bu.  =  §  of  a;,  a;  = bushels. 


PART    I.  129 


REVIEW. 


1.  From  April  25  th  to  May  5  th  it  is  days. 

(a)  How  many  days  from  April  25th  to  August  5  th  ? 

2.  From  Feb.  20th  of  a  leap-year  to  March  10th,  it  is 
days. 

(b)  How  many  days  from  Feb.  20th  of  a  leap-y^ar  to  July 
4th? 

(c)  Divide  64  by  f .     (d)  Multiply  64  by  |. 
Compare  the  answers  to  problems  (c)  and  (d). 

3.  ^  = hundredths.        ^  = thousandths. 

(e)  Change  |  to  hundredths,    (f)  Change  -f  to  lOOOths. 
(g)  Change  -^  to  hundredths,   (h)  Change  f  to  lOOOths. 

Change  the  following  decimal  fractions  to  common  fractions  and 
reduce  them  to  their  lowest  terms : 

4.  .37|-=*       .621- =         .87^=         .16|=         .33-1-  = 
(i)  .15  =       (j)  .85  =      (k)  .65  =      (l)  .75  =      (m)  .55  = 

Find  the  cost : 

5.  5500  brick  at  $8  per  M.      6.  250  brick  @  $8. 
(n)  7600  brick  at  $6  per  M.     (o)  520  brick  @  $6. 

(p)  There  is  a  4-foot  gravel. walk  around  a  rectangular 
grass  plot  that  is  30  ft.  by  40  ft.  How  many  feet  in  the 
perimeter  of  the  grass  plot  ?  (q)  How  many  feet  in  the  per- 
imeter of  the  outside  of  the  walk  ?  (r)  How  many  square 
feet  in  the  grass  plot?  (s)  How  many  square  feet  in  the 
gravel  walk? 

(t)  If  one  ton  of  coal  is  worth  $7.50,  how  much  is  14  cwt. 
of  coal  worth  ? 

(u)  If  one  hundredweight  of  corned  beef  is  worth  $4.50, 
90  lb.  of  beef  is  worth  how  much? 

*  Divide  the  numerator  and  the  denominator  of  -^^^  by  12^. 


130  COMPLETE    ARITHMETIC. 

MISCELLANEOUS    PEOBLEMS. 

1.  I  am  thinking  of  the  surface  of  a  box  that  is  6  in. 
long,  4  in.  wide,  and  3  in.  high.     The  surface  of  the  bottom 

is square  inches.    The  surface  of  one  side  is square 

inches.     The  surface  of  one  end  is square  inches.     The 

surface  of  the  entire  box — top,  bottom,  sides,  and  ends — is 
square  inches. 

(a)  Find  the  entire  surface  of  a  box  that  is  15  in.  long, 
12  inches  wide,  and  8  inches  high. 

(b)  Make  a  receipted  bill  of  the  following  goods  sold  by 
yourself  to  your  teacher : 

March  6,  150  lb.  sugar  @  4^^;  7  lb.  coffee  @  35^; 
March  18,  4  lb.  cheese  @  15^-;  1  vinegar  barrel,  $1.00; 
March  25,  6  gal.  kerosene  @  12^;  3  gal.  syrup  @  45^. 

(c)  A  farmer  sold  15  head  of  cattle  weighing  19650  lb.  at 
$5.40  cwt.  How  much  did  he  receive  for  them  ?  (d)  What 
was  the  average  weight  per  head  ? 

(e)  If  there  is  1  half  a  ream  in  a  package  of  paper,  how 
many  sheets  in  6  such  packages  ? 

(f)  A  merchant  had  $275  in  the  bank  Monday  morning. 
His  deposits  for  the  week  were  as  follows :  Monday,  $86 ; 
Tuesday,  $55  ;  Wednesday,  $72  ;  Thursday,  $64 ;  Friday, 
$83 ;  Saturday,  $124.  He  drew  from  the  bank  during  the 
week  $354.24.     How  much  remained  on  deposit  ? 

2.  I  gave  3  doz.  eggs,  worth  15^  a  dozen,  in  part  pay- 
ment for  20  lb.  sugar  at  4^  a  pound.  There  remains  unpaid 
cents. 

(g)  I  gave  2250  lb.  hay  at  $10  per  ton  in  part  payment 
for  2  tons  of  coal  @  $6.50  per  ton.  How  much  remains 
unpaid  ? 


PART    I.  131 

SIMPLE    NUMBERS. 

1.  The  sum  of  two  numbers  is  38 ;  ona  of  the  numbers 
is  12  ;  the  other  number  is  . 

(a)  The  sum  of  two  numbers  is  12346;  one  of  the  num- 
bers is  4734.     What  is  the  other  number  ? 

2.  The  difference  of  two  numbers  is  17;  the  less  num- 
ber is  45  ;  the  greater  number  is  . 

(b)  The  difference  of  two  numbers  is  547 ;  the  less ' 
number  is  3476.     What  is  the  greater  number? 

3.  The  difference  of  two  numbers  is  14;  the  greater 
number  is  45  ;  the  less  number  is  . 

(c)  The  difference  of  two  numbers  is  607;  the  greater 
number  is  4045.     What  is  the  less  number  ? 

4.  In  a  problem  the  multiplier  is  6  and       1  n^^JJiplicand. 

^                                 ■'^  6  multiplier, 

the  product  is  $42 ;  the  multiplicand  is .  "142"  product. 

(d)  In  a  problem  the  multiplier  is  fifteen  and  the  product 
is  nine  hundred  forty-five.     What  is  the  multipHcand  ? 

5.  In   a  problem   the   multiplicand  is    ^25  multiplicand. 

$25  and  the  product  is  $125;  the  multi- ^i-r?r-  '"^^P-^®^- 

^  $125  product, 

puer  is . 

(e)  In  a  problem  the  multiplicand  is  two  hundred  thirty- 
five  dollars  and  the  product  is  five  thousand  four  hundred 
five  dollars.     What  is  the  multiplier  ? 

6.  In  a  problem  the  divisor  is  $12  and  the  quo-  $12)  ? 
tient  is  8  ;  the  dividend  is .  8 

(f)  In  a  problem  the  divisor  is  $146  and  the  quotient  is 
27.     What  is  the  dividend  ? 

7.  In  a  problem  the  quotient  is   12   and  the     $9) 
divisor  is  $9;  the  dividend  is .  12 


132  COMPLETE    ARITHMETIC. 

COMMON   FRACTIONS. 

1.  Three  fourths  of  36  feet  are  — —  feet,     f  of  37  = 

(a)  Three  fourths  of  984  feet  are  how  many  feet  ? 

(b)  Three  fourths  of  985  =        (c)  Three  fourths  986  = 

2.  36  feet  are  three  fourths  of  feet.  37  ft.  are  three 

fourths  of  and feet.     38  ft.  are  J  of  . 

(d)  576  ft.  are  three  fourths  of  how  many  feet  ? 

(e)  577  is  three  fourths  of  what?     (f)  578  is  f  of  what? 

3.  Paul  spent  ^  of  the  money  his  father  gave  him  for  a 
book  and  ^  of  it  for  a  knife,  and  had  12^  left.  Before  he 
spent  any  money  he  had  cents.* 

(g)  A  man  spent  ^  of  his  month's  wages  for  fuel  and  J-  of 
it  for  groceries,  and  had  $17.25  left.  How  much  was  his 
wages  ?  (h)  How  much  did  he  spend  for  fuel  ?  (i)  How 
much  did  he  spend  for  groceries  ? 

4.  Harris  paid  $4  for  chickens  at  f  of  a  dollar  each.     He 

bought  chickens.     4  ^  |-  means,  find  how  many  times 

2  fi/ths  are  contained  in  4  (20  fifths). 

(j)  A  man  paid  $375  for  wheat  at  $f  a  bushel.  How 
many  bushels  did  he  buy  ? 

(k)  A  man  paid  $368  for  apples  at  $lf  a  barrel.  How 
many  barrels  did  he  buy  ? 

5.  Ellis  sold  2|-  cords  of  wood  at  $4|-  a  cord ;  he  should 

receive  for  the  wood  and dollars.    (2^  times 

4|-  means,  2  times  4^  and  1  half  of  4|-.) 

(1)  A  man  sold  12^  acres  of  land  at  $56|^  an  acre.  How 
much  should  he  receive  for  the  land  ? 

*  Think  of  x  as  standing  for  the  money  his  father  gave  him.  Then  he  spent  1 
half  of  X  and  1  third  of  x,  or  5  sixths  of  x,  and  had of  x  remaining. 


PART    I.  133 


DECIMAL    FRACTIONS. 


1.  At  S60  an  acre,   3.2   acres   of  land  are  worth  

dollars.     3.2  times  60  means,  3  times  60,  plus  .2  of  60. 

(a)  At  185  an  acre,  how  much  are  6.7  acres  of  land  worth  ? 
(b)  6.2  acres  at  $75  an  acre  ? 

2.  At  $70  an  acre,  3.02  acres  of  land  are  worth  . 

3.02  times  70  means,  3  times  70,  phcs  2  hundredths  of  70. 

(c)  At  $65  an  acre,  how  much  are  5.03  acres  of  land  worth? 
(d)  8.06  acres  at  $95  an  acre? 

3.  At  $40  an  acre,  2.24  acres  of  land  are  worth  

dollars.     2.24  times  40  means,  2   times  40,  plus  2  tenths  of 
40,  plus  4  hundredths  of  40. 

(e)  At  $45  an  acre,  how  much  are  4.35  acres  of  land  worth  ? 

(f)  7.26  acres  at  $64  an  acre  ? 
Find  the  cost : 

(g)  5.34  acres  @  $265.  ^265   Price  per  acre. 

.-IN  no  A              ^  ciAat:^                         5.34    Number  of  acres, 
(h)  7.34  acres  @  $465.  

(i)    6.23  acres  @  $52.5.  $10.60    Value  of  .04  of  an  acre. 

r\    n  r\o  ^  (3>ooc;  $79.5      Value  of  .3  of  an  acre. 

0)    7.03  acres  @  $325.  ^^3^5^        Value  of  5  acres. 


(k)  3.27  acres  @  $43.5. 
(1)  5.37  acres  @  $54.6. 


$1415.10    Value  of  5.34  acres. 


(m)1.56  acres  @  $276.  note.— Whlle  the  pupil  is  multiplying  by 

(Xi)  24.3  acres  &>  S342  ^'^  separatrix  may  stand  between  the  2  and 

^    ^         '  '  6  of  the  multiplicand;  thus,  $2'' 65.    This  will 

(o)   32.6  acres  @  $41.6.  help  him  to  remember  that  he  is  really  multi- 

plying $2.65,  the  value  of  1  hundredth  of  an 
acre,  by  4.  When  he  is  ready  to  multiply  by 
3,  the  separatrix  should  be  erased  and  written  thus:  $26^5.  This  will  help  him  to  re- 
member that  he  is  really  multiplying  $26.5,  the  value  of  1  tenth  of  an  acre,  by  3. 
After  a  time  he  can  simply  imagine  the  separatrix  in  its  place.  Require  the  pupil  to 
write  the  decimal  point  in  each  partial  product  when,  in  the  process  of  multiplica- 
tion, he  reaches  the  place  where  it  belongs.  The  pupil  may  now  be  taught  that 
when  he  has  solved  a  problem  in  multiplication  of  decimals,  if  he  has  "  pointed  off" 
correctly,  the  decimal  places  in  the  product  will  he  equal  to  those  in  the  multiplicand  and 
mu>liplier  counted  together. 


134  COMPLETE    ARITHMETIC. 

DENOMINATE    NUMBERS. 

1.  From  March  26th  to  April  2d,  it  is  days.     If 

March  26th  is  Monday,  April  2d  is  . 

2.  From  April  20th  to  May  5th,  it  is  days,  or 

weeks  and  day.     If  April  20th  is  Friday,  May  5  th  is 


3.  From  April  20th  to  May  12th  it  is days,  or 

weeks  and  day.     If  April  20th  is  Friday,  May  12th  is 


(a)  How  many  days  from  April  20th  to  June  9  th  ? 

(b)  If  April  20th  is  Friday,  what  day  of  the  week  is  June 
9th  ?     (c)  June  12th  ? 

4.  In  a  year  in  which  there  is  a  Feb.  29  th,  there  are  

days,  or  weeks  and  days. 

5.  In  a  year  in  which  there  is  no  Feb.  29th,  there  are 
days,  or  weeks  and day. 

6.  If  the  first  day  of  February  of  a  common  year  is  Mon- 
day, the  first  day  of  February  of  the  next  year  is  . 

7.  If  the  first  day  of  February  of  a  leap-year  is  Monday, 
the  first  day  of  February  of  the  next  year  is  . 

8.  If  the  tenth  of  February  of  a  leap-year  is  Saturday, 
the  tenth  day  of  February  of  the  next  year  is  . 

9.  If  the  17th  day  of  April  of  a  leap-year  is  Wednesday^ 
the  17th  day  of  April  of  the  next  year  is  . 

Find  how  many  weeks  and  days : 
(d)  Apr.  10  to  July  4.*  (e)  May  5  to  July  10. 

(f)  May  18  to  Aug.  5.  (g)  June  15  to  Oct.  4. 

(h)  July  12  to  Sept.  1.  (i)  Aug.  1  to  Sept.  25. 

♦  Think  as  suggested  in  the  following ;  April  10  to  Apr.  30,  20  days;  Apr.  30  to 
May  31, 31  days  ,  May  31  to  June  30, 30  days ;  June  30  to  July  4,  4  days.  20  days  +  31 
days  +  30  days  +  4  days  =  ? 


PAKT   I. 


135 


MEASUREMENTS. 

The  following  diagram  of  a  house  and  lot  is  drawn  on  a  scale 
of  24  feet  to  an  inch. 

Norlh  side  of  Lot 


(a)  How  many  feet  long  is  the  lot,  not  including  the  walk  ? 

(b)  How  many  feet  wide  is  the  lot  ? 

(c)  How  many  feet  from  the  sidewalk  to  the  house  ? 

(d)  How  far  from  the  house  to  the  back  of  the  lot  ? 

(e)  How  far  from  the  house  to  the  north  side  of  the  lot  ? 

(f)  How  many  feet  long  is  the  house  ? 

(g)  How  many  feet  wide  is  the  front  of  the  house  ? 
(h)  How  many  feet  wide  is  the  rear  of  the  house  ? 

(i)  How  far  from  the  south  side  of  the  lot  to  the  house  ? 
(j)  How  wide  is  the  sidewalk  ?     (k)  How  much  will  the 
sidewalk  cost  at  12^  per  square  foot  ?  * 

(1)  How  much  is  the  lot  worth  at  $25  a  foot  front  ?  f 

*  There  are  306  square  feet  in  the  walk.  In  finding  the  cost  the  pupil  may  think 
that  at  1^  a  foot  it  would  cost  $3.06,  and  at  120  a  foot,  12  times  $3.06 ;  or  he  may  think 
that  if  1  foot  costs  12(?,  306  feet  would  cost  306  times  12^. 

t  The  expression  "■foot  front "  stands  for  a  strip  1  foot  wide  and  as  long  as  the  lot 
is  "deep." 


136  COMPLETE    ARITHMETIC. 

RATIO   AND   PEOPORTION. 

1.  One  ton  is  and times  800  lb.    If  800 

lb.  of  hay  is  worth  $3.00,  1  ton  is  worth  dollars. 

(a)  If  800  lb.  of  flour  is  worth  $18.50,  how  much  is  1 
ton  worth  at  the  same  rate  ? 

2.  One  mile  is  times  80  rd.      If  it  costs  $30  to 

build  80  rd.  of  fence,  to  build  one  mile  of  similar  fence  will 
cost  dollars. 

(b)  If  it  costs  $375  to  make  80  rd.  of  road,  how  much 
will  it  cost  to  make  1  mile  of  road  at  the  same  rate  ? 

3.  One  hour  is  and times  24  minutes. 

Harry  rode  5  miles  in  24  minutes ;  this  was  at  the  rate  of 
an  hour. 

(c)  A  locomotive  moved  19  miles  in  24  minutes.  This 
was  equal  to  what  rate  per  hour  ? 

4.  One  minute  is  times  20  seconds.     A  pump  rod 

made  18  strokes  in  20  seconds;  this  was  at  the  rate  of  

strokes  a  minute. 

(d)  A  certain  pulley  revolves  106  times  in  20  seconds. 
What  is  its  rate  of  revolution  per  minute  ?  (e)  How  many 
times  does  it  revolve  in  one  hour  ? 

(f)  Find  how  many  times  your  pulse  beats  in  20  seconds. 
This  is  at  what  rate  per  minute  ?     Per  hour  ? 

5.  One  pound  is  and  times  12  ounces. 

If  12  oz.  cheese  is  worth  15^,  1  lb.  is  worth  . 

(g)  If  12  ounces  of  onion  seed  is  worth  72^,  how  much  is 
1  pound  worth  ? 

6.  If  50  sq.  rods  of  land  is  worth  $20,  at  the  same  rate 
1  acre  is  worth  dollars. 

(h)  If  50  square  rods  of  land  is  worth  $345,  how  much  is 
1  acre  worth  at  the  same  rate  ? 


PART    I.  137 

PERCENTAGE. 

w 

1.  12^%  more  than  56  is .         12^%  less  than  56  = 

2.  10%  more  than  60  is .  10%  less  than  60  = 

3.  30%  more  than  50  is .  30%  less  than  50  = 

(a)  121-%  more  than  $972  =   (b)  12^-%  less  than  $972  = 
(c)  10%  more  than  $845  =      (d)  10%  less  than  $845  = 
(e)  30%  more  than  $650  =     (f)  30%  less  than  $650  = 

(^) 

4.  45  is  12|%  more  than  .*     49  is  12|-%  less  than 

-t 

5.  77  is  10%  more  than  .     54  is  10%  less  than 


6.  39  is  30%  more  than  .     28  is  30%  less  than 

(g)  $747  is  12^  per  cent  more  than  how  many  dollars  ? 
(h)  $868  is  12|-  per  cent  less  than  how  many  dollars  ? 
(i)  $572  is  10  per  cent  more  than  how  many  dollars  ? 
(j)  $729  is  10  per  cent  less  than  how  many  dollars  ? 
(k)  $299  is  30  per  cent  more  than  how  many  dollars  ? 

(5) 

7.  88  is %  more  than  80.      56  is %  less  than  64. 

8.  63  is %  more  than  56.      35  is %  less  than  50. 

(1)  $350  is  how  many  per  cent  less  than  $500  ? 

(m)  $450  is  how  many  per  cent  more  than  $400  ? 

*  Think  of  x  as  standing  for  the  number  sought.  Then  45  =  a;  and  1  eighth  of  x, 
or  I  of  X.  Since  45  is  9  eighths  of  x,  1  eighth  of  a;  is  5,  and  the  whole  of  a;  is  8  times 
5  =  40. 

1 49  =  X  less  1  eighth  of  x,  or  |  of  x.  Since  49  is  §  of  x,  1  eighth  of  x  Is  7,  and  the 
whole  of  X  is  8  times  7  =  56. 


138  COMPLETE   AKITHMETIC. 

PERCENTAGE. 

1.  Five  years  ago  the  population  of  a  certain  city  was 
6000;  it  has  increased  33-|-  %  ;  its  present  population  is  . 

(a)  Eight  years  ago  the  population  of  a  certain  city  was 
9750;  it  has  increased  33i-%.  What  is  its  present  popula- 
tion? 

•2.  A  man  sold  a  horse  for  $100  which  was  25  %  more  than 
he  gave  for  it.     He  gave  dollars  for  the  horse. 

(b)  A  man  sold  a  farm  for  $6825  which  was  25%  more 
than  he  gave  for  it.     How  much  did  he  give  for  the  farm  ? 

3.  Ahce  has  50^;  Jane  has  60^;  Jane  has  %  more 

than  Alice;  Alice  has  %  less  than  Jane.* 

(c)  Mr.  Lyon  has  600  acres  of  land;  Mr.  Whitney  has  400 
acres.  Mr.  Lyon  has  how  many  per  cent  more  than  Mr. 
Whitney  ?  (d)  Mr.  Whitney  has  how  many  per  cent  less 
than  Mr.  Lyon? 

4.  By  selling  a  horse  for  $60,  the  owner  would  lose  25%; 
the  horse  cost  dollars. 

(e)  By  selling  a  farm  for  $4320,  the  owner  would  lose 
25%.     How  much  did  the  farm  cost  him? 

5.  By  selling  a  horse  for  $60,  the  owner  would  gain 
25%;  the  horse  cost  dollars. 

(f)  By  selling  a  farm  for  $5300,  the  owner  would  gain 
25%.     How  much  did  the  farm  cost  him? 

6.  25%  of  the  vinegar  in  a  cask  leaked  out  and  36  gal. 
remained;  before  the  leakage  there  were  gallons. 

(g)  25%  of  the  water  in  a  tank  leaked  out  and  465 
gallons  remained.  How  many  gallons  in  the  tank  before  the 
leakage? 

*lu  oue  paxt  of  this  problem  50  is  the  Dase;  in  the  other  part,  60  is  the  base. 


PART    I.  139 


REVIEW. 


1.  The  sum  of  three  numbers  is  27  ;  two  of  the  numbers 
are  8  and  1 0  ;  the  other  number  is  . 

(a)  The  sum  of  three  numbers  is  2756 ;  two  of  the  num- 
bers are  784  and  975.     What  is  the  other  number? 

2.  Arthur  rode  on  his  bicycle  three  consecutive  hours ; 
the  first  hour  he  rode  12  miles;  the  second  hour,  10  miles, 
and  the  third  hour  8  miles ;  his  average  speed  per  hour 
was  miles. 

(b)  The  attendance  at  a  certain  school  for  one  week  was 
as  follows :  Monday,  35  ;  Tuesday,  38  ;  Wednesday,  37  ; 
Thursday,  36  ;  Friday,  34.  What  was  the  average  daily 
attendance  ? 

3.  At  $74  per  ton,  2  J  tons  of  coal  cost .* 

(c)  At  $348 1-  an  acre,  how  much  will  5|-  acres  cost  ? 

(d)  At  $348.50  an  acre,  how  much  will  6.5  acres  cost  ? 

(e)  Compare  the  answers  to  (c)  and  (d).  How  much  is 
their  difference  ? 

4.  The  first  day  of  January,  1897,  was  Friday.  Tell  the 
day  of  the  week  of  the  first  day  of  January  of  each  of  the 

following  years :  1898, ;  1899, ;  1900, ;  1901;t 

;  1902, ;  1903, ;  1904, ;  1905, ;  1906, 

;  1907, ;  1908, ;  1909, ;  1910, . 

(f)  Upon  what  day  of  the  week  will  the  first  day  of  Janu- 
ary, 1925,  occur? 

5.  A  plot  of  a  certain  garden  is  drawn  on  a  scale  of  20 
feet  to  an  inch.     A  line  3|  inches  long  represents  feet. 

(g)  A  certain  map  is  drawn  on  a  scale  of  25  mi.  to  an  inch. 
A  line  15^  in.  long  represents  how  many  miles  ? 

*  2  times  7J  and  1  half  of  7i. 

t  Remember  that  the  year  1900  is  not  a  leap  year. 


140  COMPLETE    ARITHMETIC. 


MISCELLANEOUS   PROBLEMS. 


1.  In  a  pane  of  glass  9  in.  by  12  in.  there  are  sq. 

inches. 

(a)  How  many  square  incnes  in  36  panes  of  glass  each  9 
in.  by  12  in.?     (b)  How  many  square  feet  ? 

2.  Mr.  Black  received  $30  per  month  as  rent  for  a  house. 
In  one  year  he  received  dollars. 

(c)  At  $35  per  month,  how  much  is  the  rent  of  a  house 
for  2  years  and  6  months  ? 

3.  I  paid  $2.00  for  coffee  at  25^  a  pound;  I  purchased 

pounds. 

(d)  Paid  $34.75  for  coffee  at  25^  a  pound.  How  many 
pounds  were  purchased  ? 

4.  In  a  floor  12  ft.  by  12  ft.  there  are  square  feet; 

there  are  square  yards. 

(e)  In  a  lot  24  feet  by  96  feet,  there  are  how  many 
square  feet  ?     (f )  How  many  square  yards  ? 

5.  If  a  train  moves  at  the  rate  of  20  miles  an  hour,  to 
move  110  miles  will  require  — —  hours. 

(g)  If  a  train  moves  at  the  rate  of  35  miles  an  hour,  how 
long  will  it  take  to  go  1000  miles? 

6.  A  boy  bought  10  chickens  for  25^  each,  and  10  for 
35^  each;  the  average  price  paid  was  •  cents. 

(h)  A  man  bought  10  horses  at  $135  each  and  10  at 
$124.50  each.     Wliat  was  the  average  price? 

7.  A  man  sold  a  horse  at  |-  of  what  it  cost  him,  thereby 
losing  $10;  the  horse  cost  him  dollars;  he  sold  it  for 

dollars. 

(i)  A  man  sold  a  farm  for  f  of  what  it  cost  him,  thereby 
losing  $1275.  How  much  did  the  farm  cost  him?  (j)  For 
how  much  did  he  sell  it  ? 


PART    I.  .141 


SIMPLE    NUMBERS. 


Review  page  11. 

1.  Name  five  odd  numbers;  five  even  numbers. 

2.  Name  three  exact  divisors  of  36;  of  48;  of  75. 

Review  page  21. 

3.  Name  five  integral  numbers ;  five  fractional  numbers ; 
five  mixed  numbers. 

4.  ^f  and  .7  are numbers.    4^  and  7.2  are . 

Review  page  31. 

5.  Name  five  prime  numbers ;  five  composite  numbers. 

6.  Which  of  the  following  are  prime  and  which  are  com 
posite?     2,  22,  5,  37,  45,  49,  53,  72,  87. 

Review  page  41. 
.7.  What  are  the  prime  factors  of  36  ?     Of  35  ?     Of  34  ? 
Of  33? 

8.  Of  what  number  are  2,  2,  3,  and  5  the  prime  factors* 

Review  page  51. 

9.  Name  three  common  multiples  of  4  and  6. 

10.  What  is  the  least  common  multiple  of  4  and  6  ? 

Review  page  61. 
(a)  Multiply  746  by  20.        (b)  Multiply  394  by  80. 
(c)  Multiply  547  by  300.      (d)  Multiply  834  by  70G. 

Review  page  71. 
(e)  Divide  891  by  30.  (f)  Divide  1265  by  50. 

(g)  Divide  728  by  40.  (h)  Divide  2478  by  70. 

Review  page  81. 
(i)  A  farmer  bought  30  sheep ;  for  5  of  them  he  paid  $6 
per  head  ;  for  1 0  he  paid  $5  per  head  ;  for  the  remainder  he 
paid  $70.     How  much   did  the   30   sheep  cost  him  ?     (j) 
What  was  the  average  price  per  head  ? 


142  COMPLETE    ARITHMETIC. 

COMMON   FRACTIONS. 

Review  page  12. 

1.  Name  three  fractions  that  have  a  common  denom- 
inator ;  three  that  do  not  have  a  common  denominator. 

2.  Change  the  following  to  equivalent  fractions  having  a 
common  denominator:  |  and  f. 

Review  page  32. 

3.  Tell  the  terms  of  each  of  the  following :     4,  |,  y^ 

4.  Keduce  each  of  the  following  to  its  lowest  terms :  ^^, 

18     15     45      27         /o\240         /K\    5  75 
TY'  YO^  ¥T'    3" 3*       K"^)    SWO'       V"/  '^JZ- 

5.  Eeduce  each  of  the  following  to  a  whole  or  mixed 

Tinmbpr-       il     25     24     32     97         /p\335         /A\    4  76 

Review  page  42. 

6.  Eeduce  each  of  the  following  to  an  improper  fraction : 
9f,7|,llf,5f     (e)28f.     (f)  47|.     (g)  94f     (h)  86^- 


Review  page  52. 

(i) 

7      1       5     _ 

(J)  tV  +  tV  = 

(k)A  +  A 

(1) 

i  -  i  = 

(m)TV-A  = 

(n)  tV  -  tV 

(o) 

|x  1  = 

(p)  A-  X  f  = 

(<1)  4  X  i 

« 

\i^  i  = 

(s)tV^tV  = 

(t)    T^TT^A 

Review  pages  62  and  72. 
(u)  Find  the  product  of  794^  multiplied  by  6^.* 
(v)  Find  the  quotient  of  835 1-  bu.  divided  by  2^  bu.* 
(w)Find  the  quotient  of  654^  bu.  divided  by  9.* 


*  Solve,  and  tell  a  suggested  number  story. 


'6.40 
3.7 


PART    I.  143 

DECIMAL    FRACTIONS. 
Review  pages  13,  23,  33,  43,  53,  63,  73,  and  133. 
Observe  again  the  fact  that  when  a  problem  in 
multiplication  of   decimals  has  been  solved  ac- 
4.480        curately,  the  number  of   decimal  places  in  the 
19.20  product  is  equal  to  the  number  of  decimal  places 

23  680        ^^  ^^^  multiplicand  and  multiplier  counted  to- 
gether.    This  fact  should  be   used   as  a  test  of 
^73.42        the  accuracy  of  the  work  rather  than  as  a  rule 

3,56        for  "pointing  off." 
AAr.rn  Observe  that  when  the  decimal  point  in  the 

Q<^7in  ^^^^   partial   product   has   been   located,  the   re- 

99rk  9A  mainder   of    the    "pointing   off"    may   be   done 

'- mechanically  by  placing  the  point  in  each  of  the 

261.3-752        other    partial    products    and    in    the    complete 
product,  directly  under  the  one  in  the  first  par- 


-005)38.455-*     iial  product. 


7691. 


Review  pages  83,  93,  and  103. 


.05)38. 4 7-51  ^^^  abstract  work  in  division  of  decimals  may 

rjnQ  r  be  regarded  as  belonging  to  Case  I. ;  that  is,  the 

pupil  may  consider  that  he  is  to  find  how  many 
5^)38  4- 75t      times  the  divisor  is  contained  in  the  dividend. 

Before  beginning  to  divide,  place  a  separa- 

76.95         trix  {^)  in  the  dividend  immediately  after  that 

figure  in  the  dividend  that  is  of  the  same  denom- 

5)oo.  47o§      ination  as  the  right  hand  figure  of  the  divisor. 

7.695  When  in  the  process  of  division  this  separatrix 

is  reached,  the  decimal  point  must  he  written  in 

.5)78.0"  II      the  quotient. 

156.  *Find  how  many  times  5  thousandths  are  contained  in 

38455  thofusandths. 

fFind  how  many  times  5  hundredths  are  contained  in 
.05)78.00  ^      3847  hundredths. 

~TTTT~  J  Find  how  many  times  5  tenths  are  contained  in  384 

15  dU.  tenths. 

§  Find  how  many  times  5  units  are  contained  in  38  units. 
j!  Find  how  many  times  5  tenths  are  contained  in  780  tenths. 
If  Find  how  many  times  5  hundredths  are  contained  in  7800  hundredths. 


144  COMPLETE    ARITHMETIC. 

DENOMINATE    NUMBERS. 
Review  page  14. 

1.  One  half  a  ton  is lb.     1  tenth  of  a  ton  = lb. 

1  hundredth  of  a  ton  = lb.     1  thousandth  of  a  ton  = 

lb. 

Review  page  24. 

2.  A  bushel  of  wheat  weighs pounds. 

(a)  72  bu.  of  wheat  weigh  how  much  more  than  2  tons  1 

Review  page  34. 

(b)  Change  3.26  tons  to  pounds. 

(c)  4.7  tons  are  how  many  pounds  ? 

(d)  Change  3264  lb.  to  tons,     (e)  5624  lb.  to  tons. 

Review  page  44. 

(f)  Find  the  cost  of  7360  lb.  coal  at  $7.25  per  ton. 

(g)  Find  the  cost  of  5360  lb.  hay  at  $9.50  per  ton. 

Review  page  54. 

(h)  Change  28  rods  to  feet,      (i)  7  miles  to  rods, 
(j)  Change  506  yd.  to  rods,     (k)  2880  rods  to  miles. 
Review  page  64. 

(1)  The  gross  weight  of  a  load  of  bran  was  2850  lb. ;  tare 
1275  lb.     Find  the  cost  at  $8  per  ton  ? 

(m)The  gross  weight  of  a  load  of  oats  was  2970  lb.;  tare 
1050  lb.  How  many  bushels  ?  (n)  Find  the  cost  at  25^  a 
bushel. 

Review  page  74. 

(o)  A  mountain  is  11000  ft.  high.  How  many  feet  more 
than  2  miles  high  is  it  ? 

(p)  A  mountain  is  5  mi.  high.    How  many  feet  high  is  it  ? 


PART   I.  145 

MEASUREMENTS. 
Review  pages  15  and  25. 

1.  Which  is  the  larger,  a  five  foot  square  or  an  oblong  4 
feet  by  6  feet  ? 

(a)  Which  is  the  larger,  a  25  ft.  square  or  an  oblong  26  ft. 
by  24  ft.? 

(b)  Find  the  area  of  a  15  ft.  square. 

(c)  Find  the  solid  content  of  a  1 5  ft.  cube. 

Review  page  35. 

2.  Every  rectangular  figure  has sides  and right 

angles.     If  the  sides  are  equal,  the  figure  is  a  .     If  two 

of  the  sides  are  longer  than  the  other  two,  the  figure  is 
an  . 

3.  Draw  a  4-sided  figure  that  is  neither  a  square  nor  an 
oblong.     Is  the  figure  you  have  drawn  rectangular  ? 

4.  All  the  angles  of  a  square  are  angles. 

5.  All  the  angles  of  an  oblong  are  angles. 

6.  Angles  that  are  not  right  angles  are  either angles 

or  angles. 

Review  page  45. 

7.  Every  rectangular  solid  has  faces.     These  faces 

may  be  either  squares  or  .     If  they  are  all  squares  the 

sohd  is  a  . 

8.  Cut  from  a  potato  or  a  turnip  a  solid  with  six  faces, 
some  of  which  .are  not  rectangular.  Observe  the  acute 
angles  and  the  obtuse  angles. 

Review  pages  55,  65,  75,  and  85. 

(d)  In  2^  cords  there  are  how  many  cubic  feet? 

(e)  In  2 1-  acres  there  are  how  many  square  rods  ? 


146  COMPLETE   ARITHMETIC. 

RATIO   AND    PROPORTION. 
Review  pages  16,  26,  36,  46,  56,  66,  76,  86,  and  96. 

1.  A  1-ft.  square  equals  what  part  of  a  2 -ft.  square  ? 

2.  A  1-ft.  cube  equals  what  part  of  a  2-ft.  cube? 

3.  A  2-ft.  square  equals  what  part  of  a  3-ft.  square  ? 

4.  A  2-ft.  cube  equals  what  part  of  a  3-ft.  cube  ? 

5.  A  2-yd.  square  equals  how  many  times  a  1-yd.  square  ? 

6.  A  2-yd.  cube  equals  how  many  times  a  1-yd.  cube  ? 

7.  A  3-yd.  sq're  equals  what  part  of  a  4-yd.  square  ? 

(a)  A  10-rod  square  equals  what  part  of  a  12-rd.  sq're  ? 

(b)  A  12-rd.  sq're  equals  what  part  of  a  16-rd.  sq're  ? 

(c)  A  12-rd.  sq're  equals  what  part  of  a  24-rd.  sq're  ? 

(d)  A  12-rd.  sq're  equals  what  part  of  a  36-rd.  sq're  ? 

8.  A  3-ft.  cube  equals  what  part  of  a  4-ft.  cube  ? 

(e)  A  3-ft.  cube  equals  what  part  of  a  5 -ft.  cube  ? 

(f)  A  3-ft.  cube  equals  what  part  of  a  6 -ft.  cube  ? 

(g)  A  3-ft.  cube  equals  what  part  of  a  9 -ft.  cube  ? 

9.  A  l-ft.  square  equals  what  part  of  a  1-ft.  square  ? 
(h)  A  ^-ft.  cube  equals  what  part  of  a  1-ft.  cube  ? 

10.  A  1-ft.  sq're  equals  how  many  times  a  ^-ft.  sq're  ? 
(i)  A  1-ft.  cube  equals  how  many  times  a  i-ft.  cube  ? 

11.  The  surface  of  a  1-foot  cube  equals  what  part  of  the 
surface  of  a  2 -foot  cube  ? 

(j)  The  surface  of  a  2-ft.  cube  equals  what  part  of  the 
surface  of  a  3-ft.  cube  ? 

(k)  The  surface  of  a  1-ft.  cube  equals  what  part  of  the 
surface  of  a  1-yd.  cube  ? 

(1)  If  a  1-inch  cube  of  silver  is  worth  $3.60,  how  much 
is  a  3 -inch  cube  of  the  same  metal  worth  ? 


PART    I.  147 

PERCENTAGE. 
Review  pages  17  and  18. 

1.  25%  of  24  is  . .        (a)  Find  25%  of  $3479. 

2.  24  is  25%  of  .        (b)  S3479  is  25%  of  what? 

3.  8  is  %  of  24.  (c)  75  is  what  %  of  375  ? 

Review  pages  27  and  28. 

4.  121-%  of  72  is  .      (d)  Find  12J%  of  $650. 

5.  72  is  12%  of  .        (e)  $650  is  12i-%  of  what? 

6.  12  is  %  of  72.        (f)   35  is  what  %  of  245  ? 

Review  pages  37  and  38. 

7.  10%  of  45  is  .        (g)  Find  10%  of  $725. 

8.  45  is  10%  of  .        (h)  $725  is  10%  of  what  ? 

9.  5  is  %  of  45.  (i)   55  is  what  %  of  440  ? 

Review  pages  47  and  48. 

10.  66|%  of  48  is  .      (j)  Find  66f  of  $756. 

11.  48  is  66f%  of  .      (k)  $756  is  66|%  of  what? 

12.  36  is  %  of  48.        (1)  $450  is  what  %  of  $600  ? 

Review  pages  57  and  58. 

13.  60%  of  75  is  .        (m)Find  60%  of  $810. 

14.  75  is  60%  of  .        (n)  $810  is  60%  of  what? 

15.  20  is  %  of  40.        (o)  $84  is  what  %  of  $210  ? 

Review  pages  67  and  68. 

16.  80%  of  60  is  .        (p)  Find  80%  of  $640. 

17.  60  is  80%  of  .        (q)  $640  is  80%  of  what? 

18.  50  is  %  of  60.        (r)  $550  is  what  %  of  $660  ? 

Review  pages  77,  78,  87,  and  88. 

19.  37|^%  of  24  is  .      (s)  Find  37^%  of  $576. 

20.  24  is  37^%  of .       (t)  $576  is  37^%  of  what  ? 

21.  24  is  %  of  80.        (u)  $675  is  what  %  of  $750  ? 


148  COMPLETE    ARITHMETIC. 

PERCENTAGE. 
Review  pages  97  and  98. 

1.  One  %  of  357  =  (a)  Find  13%  of  357. 

2.  39  is  3%  of .  (b)  264  is  8%  of  what  ? 

3.  15  is %  of  300.*         (c)  60  is  what  %  of  750  ?t 

Review  pages  107  and  108. 

4.  One  %  of  736  =  (d)  Find  8  J%  of  736. 

5.  108  is  9%  of .  (e)  375  is  5%  of  what  ? 

6.  57  is  %  of  300.  (f)  41.5  is  what  %  of  830? 

Review  pages  117  and  118. 

7.  25%  more  than  80  =  25%  less  than  80  = 

8.  40  is  25  %  more  than 4    45  is  25  %  less  than . 

9.  75  is %  more  than  60.§     60  is %  less  than  75. 

10.  AUce  has  $40  ;  Jane  has  $50  ;  Mary  has  $60. 
(g)  Jane  has  what  per  cent  more  than  Ahce  ? 
(h)  Mary  has  what  per  cent  more  than  Jane  ? 
(i)  Mary  has  what  per  cent  more  than  Alice  ? 
(j)  Jane  has  what  per  cent  less  than  Mary  ? 
(k)  Alice  has  what  per  cent  less  than  Jane  ? 
(1)  Alice  has  what  per  cent  less  than  Mary  ? 
(m)  Alice's  money  equals  what  %  of  Jane's  money  ? 
(n)  Alice's  money  equals  what  %  of  Mary's  money  ? 
(o)  Jane's  money  equals  what  %  of  AUce's  money  ? 
(p)  Jane's  money  equals  what  %  of  Mary's  money  ? 
(q)  Mary's  money  equals  what  %  of  Alice's  money  ? 
(r)  Mary's  money  equals  what  %  of  Jane's  money  ? 

*  First  find  1%  of  300. 

t  First  find  l5i  of  750. 

$  Let  a;  =  the  number  sought,  the  base ;  then  40  =  a;  and  1  fourth  of  x,  or  I  of  z. 

I  75  is  how  many  more  than  60  ?    15  is  what  ji  of  60  ? 


PART    1.  149 

Review  pages  113  and  123. 

1'  i= hundredths,     (a)  -J  = thousandths. 

(b)  Change  .275  to  a  common  fraction  and  reduce  it  to  its 
lowest  terms.       (c)  .375.       (d)  .425.       (e)  .575.       (f)  .625. 

Review  pages  133  and  143. 

(g)  Find  the  cost  of  6.28  acres  of  land  at  $2.75  an  A.* 
(h)  Find  the  cost  of  3.46  tons  of  coal  @  $6.75  a  ton. 

i)   Divide  6.25  by  5.     (6.^25  -^  5  units.) 

j)   Divide  6.25  by  .5     (6.2^5  ^  5  tenths.) 

k)  Divide  6.25  by  .05.     (6.25V  5  hundredths.) 

1)   Divide  36  by  5.     (36'  divided  by  5  units.) 

m)  Divide  36  by  .5.     (36.0'  -^  5  tenths.) 

n)  Divide  36  by  .05.     (36.00'  ^  5  hundredths.) 

o)  Divide  36  by  .005.     (36.000'  -^  5  thousandths.) 

p)  Divide  57.26  by  7.     (57.'26  divided  by  7  units.) 

q)  Divide  57.26  by  .7.     (57.2'6  h-  7  tenths.) 

r)   Divide  57.26  by  .07.     (57.26'  -^  7  hundredths.) 

s)  Divide  57.26  by  .007.     (57.260'  -^  7  thousandths.) 

t)   Divide  67.5  by  25.     (67.'5  divided  by  25  units.) 

u)  Divide  67.5  by  2.5.     (67.5'  -^  25  tenths.) 

v)  Divide  67.5  by  .25.     (67.50'  -4-  25  hundredths.) 

w)  Divide  67.5  by  .025.     (67.500'  -^  25  thousandths.) 

x)  Divide  6.75  by  25.     (25  units  in  6  units  =  0.,  etc.) 

y)  Divide  6.75  by  2.5.     (6.7'5  -^  25  tenths.) 

*  Reqxiire  the  pupil  to  put  the  work  on  the  blackboard  and  to  explain  by  telling 
(1)  the  cost  of  1  hundredth  of  an  acre;  (2)  of  8  hundredths;  (3)  of  1  tenth;  (4)  of  2 
tenths;  (5)  of  1  acre;  (6)  of  6  acres;  (7)  of  6.28  acres.  How  many  decimal  places  in 
the  product?    How  many  in  the  multiplicand  ?    How  many  in  the  multiplier  ? 


CONTENTS— PART  II. 


Pages 
Notation,  -  .  .  .  .  .  .     I5i_i58 

Addition, .  161-168 

Subtraction,     ---._..     171-178 
Multiplication,      -  -  -  -  .  .  181-188 

Division,  ----...     I9i_i98 

Properties  of  Numbers,  -  -  .  _  201-206 

Divisibility  of  Numbers,      -  -  -  _  .      211-216 

Fractions,   -  -  -  221-228,231-238,241-248,251-256 

Percentage, 261-266,271-276 

Discounting  Bills,     -----  281 

Discounts  from  List  Price,      -  -  -  -  282 

Selling  on  Commission,        -  -  -  -  283 

Taxes,  ----...  284 

Insurance,        --....  285 

Interest,     -  -  -  -  .  ...      291-296 

Promissory  Notes,     -  -  -  .  .  301-306 

Stocks  and  Bonds,  -  -  -  .  _     311-316 

Ratio  and  Proportion,     -  .  .  .        321-328,  a31-338 

Powers  and  Eoots,     -----  341-348,  351-358 

Metric  System,       ------  361-368 

Algebra,  -  -  -  157, 158;  167, 168;  177, 178,  etc. 

Geometry,    -  -  -  .        159, 169, 179, 189, 199,  209,  etc. 

Miscellaneous  Problems,     -  160, 170, 180, 190,  200,  210,  etc. 


150 


PART   II. 

NOTATION. 

1.  The  expression  of  numbers  by  symbols  is  called 
notation. 

2.  In  mathematics  two  sets  of  symbols  are  employed  to 
represent  numbers ;  namely,  ten  characters — 1,  2,  3,  4,  5,  6, 
7,  8,  9,  0 — called  figures ;  and  the  letters  a,  h,  c,  d,  .  .  . 
X,  y,  z. 

Note. — The  figures  from  1  to  9  are  called  digits.  The  term 
significant  figures  is  sometimes  applied  to  the  digits.  The  tenth 
character  (0)  is  called  a  cipher,  zero,  or  naught. 

THE    ARABIC    NOTATION. 

3.  The  method  of  representing  numbers  by  figures  and 
places  is  called  the  Arabic  Notation.  It  is  the  principle  of 
'position  in  writing  numbers  that  gives  to  the  system  its 
great  value. 

s   ,' 

IS   I         III 


u 


c3 


+3        02       ^      ^        02 


oaJo          ?5.SS  (Da)a>.-ga)a<x> 

^T^^^oj'^rS  z^   ^^    ^         ^    ^-^   ^ 

^    ^^     Ph-S    ^2  OOO^OOO 

1  1    S  ^  1^    §  1  .^  .^  .^  I  .-§  .^  .-§ 
S*r^"^'-i^^lrf  d    G    ^  -^    iz:    c    a 

2  4  3  8.596  2438.596 

151 


152  COMPLETE    ARITHMETIC. 

4.  A  figure  standing  alone  or  in  the  first  place  represents 
primary  units,  or  units  of  the  first  order;  a  figure  standing 
in  the  second  place  represents  units  of  the  second  order;  a 
figure  standing  in  the  third  place  represents  units  of  the 
third  order ;  a  figure  standing  in  the  first  decimal  place  rep- 
resents units  of  the  first  decimal  order,  etc. 

5.  The  following  are  the  names  of  the  units  of  eight 
orders : 

Fourth  decimal  order     .     .     .  ten-thousandths. 

Third  decimal  order       .     .     .  thousandths. 

Second  decimal  order     .     .     .  hundredths'. 

First  decimal  order    ....  tenths. 

DECIMAL   POINT. 

First  order  primary  units. 

Second  order tens. 

Third  order hundreds. 

Fourth  order thousands. 

6.  In  a  row  of  figures  representing  a  number  (342.65), 
the  figure  on  the  right  represents  the  lowest  order  given ; 
the  figure  on  the  left,  the  highest  order  given.  In  general, 
any  figure  represents  an  order  of  units  higher  than  the  figure 
on  its  right  (if  there  be  one),  and  lower  than  the  figure  on 
its  left  (if  there  be  one). 

7.  Ten  units  of  any  order  equal  one  unit  of  the  next 
higher  order;  thus,  ten  hundredths  equal  one  tenth;  ten 
tenths  equal  one  primary  unit,  etc. 

8.  The  naught,  or  zero,  is  used  to  mark  vacant  places; 
thus,  the  figures  205  represent  2  hundred,  no  tens,  and  5 
primary  units. 


PART   II.  153 

Note  1. — Observe  that  a  figure  always  stands  for  units.  If  it 
occupies  the  first  place,  it  stands  for  primary  units ;  if  it  occupies 
the  second  place,  it  stands  for  tens  (that  is,  units  of  tens);  the  third 
place,  for  hundreds  ;  the  first  decimal  place,  for  tenths ;  the  second 
decimal  place,  for  hundredths,  etc.  Thus,  a  figure  5  always  stands 
for  five — jive  primary  units,  jive  thousand,  jive  hundredths,  jive 
tenths,  according  to  the  place  it  occupies. 

Note  2. — In  reading  integral  numbers,  the  primary  unit  should 
be,  and  usually  is,  most  prominent  in  consciousness.  Thus,  the 
number  275  is  made  up  of  2  hundreds,  7  tens,  and  5  primary  units ; 
but  2  hundreds  equal  two  hundred  (200)  primary  units,  and  seven 
tens  equal  seventy  (70)  primary  units ;  these  (200  +  70  +  5)  we 
almost  unconsciously  combine  in  our  thought,  and  that  which  is 
present  in  consciousness  is  275  primary  units.  So  in  the  number 
125,246,  there  are  units  of  six  orders,  which  we  reduce  in  thought 
to  primary  units,  and  say,  one  hundred  twenty-five  thousand  two 
hundred  forty-six  primary  units. 

Note  3. — In  reading  decimals,  too,  the  primary  unit  should  be 
prominent  in  consciousness.  Thus,  .256  is  made  up  of  2  tenths,  5 
hundredths,  and  6  thousandths ;  but  2  tenths  equal  200  thousandths, 
and  5  hundredths  equal  50  thousandths ;  these  (200  +  50  +  6)  we 
combine  in  our  thought,  and  that  which  should  be  present  in  con- 
sciousness is  256  thousandths  of  a  primary  unit. 

9.  Exercise. 

Write  in  figures : 

1.  Two  hundred  fifty-four  thousand  one  hundred. 

2.  One  hundred  seventy-five  and  two  hundred  six  thou- 
sandths. 

3.  Eighty-four  and  three  hundred  five  thousandths. 

4.  Three  hundred  seven  and  eighty-seven  hundredths. 

5.  Seven  thousand  four  hundred  twenty-four. 

6.  Twenty-four  thousand  six  hundred  fifty-one. 

7.  One  hundred  thirty-five  thousand  two  hundred, 
(a)  Fiifid  the  sum  of  the  seven  members. 


154  COMPLETE    ARITHMETIC. 

10.  Exercise. 

Read  in  two  ways  as  suggested  in  the  following : 

324.61.     (1)   3   hundreds,   2   tens,  4   primary  units,  6 

tenths,  1  hundredth.     (2)  Three  hundred  twenty-four  and 

sixty-one  hundredths. 

Use  the  word  and  in  place  of  the  decimal  point  only. 

1.  2746.2.  5.  2651.4. 

2.  546.85.  6.  80.062. 

3.  24.006.  7.  2085.7. 

4.  1.6285.  8.  120.08. 

11.  Exercise. 

Observe  that  any  number  may  be  read  by  giving  the  name  of 
the  units  denoted  by  the  right-hand  figure  to  the  entire  number ; 
thus,  146  is  146  primary  units ;  21.8  is  218  tenths ;  3.25  is  325  hun- 
dredths. 

1.  27  =  2  tens  +  7  primary  units  =  27  primary  units. 

2.  2.7  =  2  primary  units  +  7  tenths  = tenths. 

3.  .27  =  2  tenths  +  7  hundredths  = hundredths. 

4.  .027  =  2  hundredths  +  7  thousandths  = thou- 
sandths. 

5.  .436  =  4  tenths  +  3  hundredths  +  6  thousandths  = 
thousandths. 

6.  5.247  =  5  primary  units  +  2  tenths  +  hundredths  -}- 
7  thousandths  =  5247 ths. 

7.  3.24      = hundredths. 

8.  5.206    = thousandths. 

9.  25.13    =  hundredths. 

10.  14.157  = thousandths. 

11.  275.4    = tenths. 

Note. — Exercise  11  and  Exercise  12  are  important  as  a  prepar- 
ation for  the  clear  understanding  of  division  of  decimals. 


PART   II.  155 

12.  Exercise. 

Observe  that  any  part  of  a  number  may  be  read  by  giving  the 
name  of  the  units  denoted  by  the  last  figure  of  the  part  to  the  entire 
part;  thus,  24.65  is  246  tenths  and  5  hundredths;  14.275  is  1427 
hundredths  and  5  thousandths.  In  a  similar  manner  read  each  of 
the  following : 

1.  2.75      = tenths  and  hnndredths. 

2.  32.46    = •  tenths  and  hundredths. 

3.  1.425    = hundredths  and  ■ thousandths. 

4.  24.596  = tenths  and  thousandths. 

5.  321.45  = tenths  and  hundredths. 

6.  14.627  = hundredths  and  thousandths. 

7.  2.6548  = hundredths  and  ten-thousandths. 

13.  Exercise. 

Observe  that  in  reading  a  mixed  decimal  in  the  usual  way,  we 
divide  it  into  two  parts  and  give  the  name  of  the  units  denoted  by 
the  last  figure  of  each  part  to  each  part ;  thus,  2346.158  is  read  2346 
(primary  units)  and  158  thousandths. 

Read  the  following  in  the  usual  manner.  Do  not  use  the  word 
and  in  reading  the  numbers  in  the  second  column  : 


1. 

200.006. 

.206. 

6. 

800  and  24. 

824. 

2. 

400.0005. 

.0405. 

7 

9000  and  6. 

9006. 

3. 

500.025. 

.525. 

8. 

2400  and  8. 

2408. 

4. 

200  and  40. 

240. 

9. 

17000  and  4. 

17004. 

5. 

700  and  35. 

735. 

10. 

46500  and  40. 

46540. 

14.  Exercise. 

Write  in  figures : 

1.  Two  hundred  and  eight  thousandths. 

2.  Two  hundred  eight  thousandths. 

3.  Six  hundred  and  twelve  thousandths. 

4.  Six  hundred  twelve  thousandths. 


156  ^  COMPLETE    ARITHMETIC. 

15.  Reference  Table. 


1 

o 
'A 

I 

a 
o 

1 

xh 

1 

"3 

a 

a 
.2 

1 

.2 

o 

9 

O 

H 

5 

157,896,275,832,456,297,143,215,367,291,326,415. 

16.  Note  the  number  of  decimal  places  in  each  of  the 
following  expressions : 

1.  .4  =  4  tenths.     (1  decimal  place.) 

2.  .27  =  27  hundredths.     (2  decimal  places.) 

3.  .346  =  346  thousandths.     (3  decimal  places.) 

4.  .2758  =  2758  ten-thousandths. 

5.  .07286  =  7286  hundred  thousandths. 

6.  .000896  =  896  millionths.     (6  decimal  places.) 

7.  .000,468,275  = billionths.      (9  decimal  places.) 

8.  .000,000,000,462  = trillionths. 

9.  .000,000,000,000,527  = quadrillionths. 

10.  In  any  number  of  thousandths  there  are decimal 

places. 

11.  In  any  number  of  millionths  there  are  decimal 

places. 

12.  In  any  number  of  billionths  there  are  decimal 

places. 

13.  In  any  number  of  hundredths  there  are  decimal 

places. 

14.  In  any  number  of   ten-thousandths    there  are   

decimal  places. 

15.  In  any  number  of   hundred  thousandths  there  are 
decimal  places. 


PAKT   II.  167 

Algebra— Notation. 

17.  Letters  are  used  to  represent  numbers ;  thus,  the  let- 
ter a,  h,  or  c  may  represent  a  number  to  which  any  value 
may  be  given. 

18.  Known  numbers,  or  those  that  may  be  known  with- 
out solving  a  problem,  when  not  expressed  by  figures,  are 
usually  represented  by  the  first  letters  of  the  alphabet ;  as, 
a,  h,  c,  d. 

Illustrations. 

(a)  To  find  the  perimeter  of  a  square  when  its  side  is  given. 

Let  a  =  one  side.* 
Then  4  a  =  the  perimeter. 
Hence  the  rule  :     To  find  the  perimeter  of  a  square,  multiply  the 
number  denoting  the  length  of  its  side  by  4. 

(b)  To  find  the  perimeter  of  an  oblong  when  its  length  and 
breadth  are  given. 

Let  a  =  the  length. 
Let  b  -  the  breadth. 
Then  2a  +  26,  or  (a  +  6)  X  2  =  the  perimeter. 
Hence  the  rule :     To  find  the  perimeter  of  an  oblong,  multiply 
the  sum  of  the  numbers  denoting  its  length  and  breadth  by  2. 

19.  Unknown  numbers,  or  those  which  are  to  be  found 
by  the  solution  of  a  problem,  are  usually  represented  by  the 
last  letters  of  the  alphabet ;  as  x,  y,  z. 

Illustration. 
(a)  There  are  two  numbers  whose  sum  is  48,  and  the  second  is 
three  times  the  first.     What  are  the  numbers  ? 
Let  X  =  the  first  number. 

Then  3  a;  =  the  second  number, 

and        a;  +  3  £c  =  48. 
4  aj  =  48. 
X  =  \2.     3  a;  =  36. 

*  That  is,  the  number  of  units  in  one  side.    The  letter  stands  for  the  number. 


158  COMPLETE    ARITHMETIC. 

20.  The  sign  of  multiplication  is  usually  omitted  between 
two  letters  representing  numbers,  and  between  figures  and 
letters ;  thus,  a  xi,  is  usually  written  ab ;  b  x  4,  is  written 
4:  b.     6  ab,  means,  6  times  a  times  b,  or  6  x  a  x  b. 

21.  Exercise. 

Find  the  numerical  value  of  each  of  the  following  expressions,  if 
a  =  8,  b  =  5,  and  c  =  2  : 

1.  <x  +  &  +  c  =  5.  2ab  = 

2.  a^b  —  c=  6.  Sabc  = 

3.  2a  +  b  +  c=  7.  2ab  +  5c  = 
4:.  a-\-b-2c=  S.  ab-\-bG  = 

(a)  Find  the  sum  of  the  eight  results. 

22.  Exercise. 

Find  the  numerical  value  of  each  of  the  following  expressions  if 
a  =  20,b  =  o,  and  c  =  2  : 

1.  3(a-\-b)  =  *  b.  a-^b  = 

2.  2(a-b)=:  6.  (a  +  &)^c  =  t 

3.  4(a  +  6  +  c)=:  7.  {a-\-b)^3c  = 

4.  2(a-{-b-c)=  8.  (a  +  2b)  ^2c  = 

(b)  Find  the  sum  of  the  eight  results. 

23.  Exercise. 

1.  If  2x  =  20,  2.  If  bx  =  40,  3.  If  6x  =  72, 

a;  =  ?  i»  =  ?  x=  1 

4.  U2x-\-Sx  =  60,  5.  If  3^  +  4^  =  56, 

'    x='i.  x  =  'i 

*  This  means,  3  times  the  sum  of  a  and  6. 

t  This  means,  the  sum  of  a  and  &,  divided  by  c. 


PART   II. 


159 


Geometry. 


24.  A  geometrical  line  has  length,  but  neither  breadth  nor 
thickness. 

XoTE. — Lines  drawn  upon  paper  or  upon  the  blackboard  are  not 
geometrical  lines,  since  they  have  breadth  and  thickness.  They 
represent  geometrical  lines. 

25.  A  straight  line  is  the  shortest  distance  from  one  point 
to  another  point. 

26.  A  curved  line  changes  its  direction  at  every  point. 

27.  A  broken  line  is  not  straight,  but  is  made  up  of 
straight  lines. 

1.  The  line  AB  is  a . 

2.  The  line  CD  is  a . 

3.  The  line  ^i^is  a . 

4.  The  line  FG  is  a . 

5.  The  Hne  JEK  is  a . 


6.  The  perimeter  of  a  square  is  a 
-  equal  lines. 


line  made  up  of 


7.  The  perimeter  of  a  regular  pentagon  is  a 
made  up  of  equal  lines. 


hne 


8.  The  circumference  of  a  circle  is  a 


line,  every 


point  in  which  is  equally  distant  from  a  point  called  the 
center  of  the  circle. 

9.  Imagine  a  straight  Hne  drawn  upon  the  surface  of  a 
stovepipe.  Can  you  draw  a  straight  line  upon  the  surface 
of  a  sphere  ? 


160  COMPLETE   ARITHMETIC. 

28.  Miscellaneous  Review. 

1.  If  a  equals  one  side*  of  a  regular  pentagon,  the  peri- 
meter of  the  pentagon  is . 

2.  li  b  equals  the  perimeter  of  a  square,  the  side  of  the 
square  equals  h  -i . 

3.  If  a  equals  a  straight  line  connecting  two  points  and 
b  equals  a  curved  line  connecting  the  same  points,  then  a  is 
than  b. 

4.  Find  the  difference  between  two  hundred  seven  thou- 
sandths, and  two  hundred  and  seven  thousandths. 

5.  How  many  zeros  in  1  million  expressed  by  figures  ? 
1  billion?     1  trillion? 

6.  How  many  decimal  places  in  any  number  of  million- 
ths?     billionths?     trillionths? 

7.  How  many  decimal  places  in  25  thousandths  ?  in  275 
thousandths  ?  in  4346  thousandths  ? 

8.  A  figure  in  the  second  integral  place  represents  units 
how  many  times  as  great  as  those  represented  by  a  figure  in 
the  second  decimal  place  ? 

9.  li  a  =  6,  b  =  2,  and  ^  =  8,  what  is  the  numerical 
value  of  the  following  ?  12  a  -\-  Sb  —  6  d. 

10.  John  had  a  certain  amount  of  money  and  James  had 
5  times  as  much;  together  they  had  354  dollars.  How 
many  dollars  had  each  ? 

Let  X  =  the  number  of  dollars  John  had. 

Then        6  x  =  the  number  of  dollars  James  had, 
and  X  -\-  5  X  =  354  dollars. 
6  a;  =  354  dollars. 
X  =  'i     5x  =  ? 

*The  expression  "  a  equals  one  side  "  means  that  a  equals  the  number  of  units 
in  one  side.  Remember  that  in  this  kind  of  notation  the  letters  employed  stand  Jot 
numbers. 


ADDITION. 

29.  Addition  (in  arithmetic)  is  the  process  of  combining 
two  or  more  numbers  into  one  number. 

XoTE  1. — The  word  number,  as  here  used,  stands  for  measured 
magnitude,  or  number  of  things. 

30.  The  sum  is  the  number  obtained  by  adding. 

31.  The  addends  are  the  numbers  to  be  added. 

32.  The  sign,  +,  which  is  read  plus,  indicates  that  the 
numbers  between  which  it  is  placed  are  to  be  added ;  thus, 
6  +  4,  means  that  4  is  to  be  added  to  6. 

33.  The  sign,  =,  which  is  read  equal  or  equals,  indicates 
that  that  which  is  on  the  left  of  the  sign,  equals  that  which 
is  on  the  right  of  the  sign ;  thus,  3+4  =  7.  5+4  +  2  = 
6  +  5. 

34.  Principles. 

1.  Only  like  numbers  can  he  added. 

2.  The  denmnination  of  the  sum  is  the  same  as  that  of 
the  addends. 

35.  Primary  Facts  of  Addition. 

There  are  forty-five  primary  facts  of  addition.  See  Elementary 
Book,  pp.  32  and  82.  The  nine  which  many  pupils  fail  to  mem- 
orize perfectly  are  given  below.     (Note  i,  p.  443.) 


7 

8 

9 

8 

9 

8 

9 

9 

9 

6 

5 

4 

6 

5 

7 

6 

7 

8 

13. 

13 

13 

14 

14 
161 

15 

15 

16 

17 

162  COMPLETE    ARITHMETIC, 

36.  Examples  of  Addition. 


1. 

2. 

3. 

3754 

37.426 

$24,305 

2862 

1.48 

$6,752 

1457 

375.062 

$375.08 

8073  413.968  $406,137 


4. 

5. 

6. 

275  acres. 

43  gal. 

2  qt. 

1  pt. 

5a-\-2b 

146  acres. 

24  gal. 

1  qt. 

1  pt. 

27a +  36 

27  acres. 

63  gal. 

3  qt. 

1  pt. 

4:6a -{-U 

448  acres. 

131  gal. 

3  qt. 

1  pt. 

78a  +  9b 

37.  Observe  that  in  written  problems  in  addition  the 
figures  that  stand  for  units  of  the  same  order  are  usually 
written  in  the  same  column. 

1.  In  example   2,  what  figures  represent  units  of  the 
second  decimal  order  ?     Of  the  third  decimal  order  ? 

2.  In  example  5,  what  figures  represent  units  of  the  first 
integral  order  ? 

38.  Observe  that  in  written  problems  in  addition  of 
denominate  numbers,  the  figures  that  stand  for  units  of  the 
same  denomination  and  order  are  usually  written  in  the 
same  column. 

1.  In  example  5,  what  figures  represent  units  of  gallons  ? 
Of  quarts  ? 

2.  In  example  5,  what  figures  represent  tens  of  gallons  ? 

3.  In  example  4,  what  figures  represent  hundreds  of 
acres  ? 


PART    II.  163 

Addition— Simple  Numbers. 

39.  Find  the  sum  of  275,  436,  and  821. 

Operation.  Explanation. 

^rj^  The  sum  of  the  units  of  the  first  order  is  12;  this  is 

^o/?  equal  to  one  unit  of  the  second  order  and  2  units  of  the 

Q^iy^  first  order.     "Write  the  2  units  of  the  first  order,  and 

add  the  1  unit  of  the  second  order  to  the  other  units  of 


1532        the  second  order. 

The  sum  of  the  units  of  the  second  order  is  13 ;  this 
is  equal  to  1  unit  of  the  third  order  and  3  units  of  the  second  order. 
Write  the  3  units  of  the  second  order,  and  add  the  1  unit  of  the 
third  order  to  the  other  units  of  the  third  order. 

The  sum  of  the  units  of  the  third  order  is  15;  this  is  equal  to  1 
unit  of  the  fourth  order  and  5  units  of  the  third  order,  each  of 
whicli  is  written  in  its  place. 

The  sum  of  275,  436,  and  821  is  1532. 

40.  Problems. 

1.  Add  3465,  4268,  3279,  6534,  5731. 

2.  Add  5732,  6721,  3466,  4269,  6535. 

3.  Add  2768,  5329,  4685,  3752,  8467. 

4.  Add  4671,  5315,  6248,  1533,  7232. 

5.  Add  375,  506,  258,  327,  580,  648,  846. 

6.  Add  436,  307,  449,  498,  736,  274,  888. 

7.  Add  625,  494,  742,  673,  574,  654,  638. 

8.  Add  564,  693,  684,  502,  376,  726,  877. 

(a)  Find  the  sum  of  the  eight  sums. 

To  THE  Teacher — Impress  upon  the  pupil  the  fact  that  in 
arithmetic  nothing  short  of  accuracy  is  commendable.  One  figure 
wrong  in  one  problem  in  ten  is  failure.  The  young  man  or  the 
young  woman  who  cannot  solve  ten  problems  like  those  on  this  page 
without  an  error,  is  worthless  as  an  accountant. 


164  COMPLETE    ARITHMETIC. 

Addition— Decimals. 

41.  Find  the  sum  of  4.327,  8.29  and  .836. 

Operation.  Explanation. 

Ac>nn  The  sum  of  the  units  of  the  third  decimal  order  is 

q\q  13  ;  this  is  equal  to  1  unit  of  the  second  decimal  order 

\oa       ^i^d  3  units  of  the  third  decimal  order.     Write  the  3 

! units  of  the  third  decimal  order  and  add  the  1  unit  of 

13.453       the  second  decimal  order  to  the  other  units  of  that 
order. 

The  sum  of  the  units  of  the  second  decimal  order  is  15 ;  this  is 
equal  to  1  unit  of  the  first  decimal  order  and  5  units  of  the  second 
decimal  order.  Write  the  5  units  of  the  second  decimal  order  and  add 
the  1  unit  of  the  first  decimal  order  to  the  other  units  of  that  order. 

The  sum  of  the  units  of  the  first  decimal  order  is  14;  this  is  equal 
to  1  unit  of  the  first  integral  order  and  4  units  of  the  first  decimal 
order.  Write  the  4  units  of  the  first  decimal  order  and  add  the  1 
unit  of  the  first  integral  order  to  the  other  units  of  that  order. 

The  sum  of  the  units  of  the  first  integral  order  is  13;  this  is 
equal  to  1  unit  of  the  second  integral  order  and  3  units  of  the  first 
integral  order,  each  of  which  is  written  in  its  place. 

The  sum  of  4.327,  8.29,  and  .836  is  13.453. 

42.  Pkoblems. 

1.  Add  474.36,  21.37,  38.007,  and  487. 

2.  Add  78.63,  61.993,  .725,  and  724.64. 

3.  Add  .7,  .84,  .375,  .0275,  and  .25326. 

4.  Add  85.997,  47.9994,  72,  and  53.93. 

5.  Find  the  sum  of  seven  hundred  ninety-eight  and  nine 
hundred  ninety-four  thousandths,  and  seven  hundred  ninety- 
four  thousandths. 

(a)  Find  the  sum  of  the  five  sums. 

To  THE  Pupil. — Can  you  solve  these  five  problems  and  find  the 
sum  of  the  five  sums  on  first  trial  without  an  error  ? 


PART    II. 


165 


Addition— United  States  Money. 

43.  Find  the  sum  of  the  money  represented  in  the  follow- 


ing columns : 

$32445 

28.47 

375.28 

6.94 

175.89 

27.56 
475.39 
802.21 
354.48 
916.37 
144.50 

75.34 


Explanation. 
Sums  of  the  units  of  each  order 


81 
70 
82 
56 
39 


246.25 
$3962.01 


Second  decimal  order  (cents) 
Fhst  decimal  order  (dimes)  . 
First  integral  order  .... 
Second  integral  order    . 
Third  integral  order      .     .     . 

Observe  that  the  8  of  the  first  sum  is  included  in  the 
70  of  the  second  sum;  that  the  7  of  the  second  sum  is 
included  in  the  82  of  the  third  sum;  that  the  8  of  the 
third  sum  is  included  in  the  56  of  the  fourth  sum,  and 
that  the  5  of  the  fourth  sum  is  included  in  the  39  of 
the  fifth  sum.  Hence  the  sum  of  the  five  sums  is  rep- 
resented by  the  figures  8962.01. 


44.  Problems.* 


1.  2.  3. 

$256.35'  $275  $725 

145.24  146  854 

321.75  281  719 

286.44  675  697 

308.92  284  716 

244.31  552  448 

986.24  496  504 

275.46  628  715 

383.27  682  603 
(a)  Find  the  sum  of  the  four  sums. 


4. 

$743.65 
854.76 
678.25 
713.56 
843.18 
769.45 
724.54 
616.73 
847.90 


*  To  THE  Pupil  —Remember  that  nothing  short  of  absolute  accuracy  is  of  any 
value  in  such  work  as  this. 


166 


COMPLETE    ARITHMETIC. 


Addition— Denominate  Numbers. 

45.  Find  the  sum  of  7  bu.  2  pk.  5  qt.,  3  bu.  3  pk.  3  qt., 
6  bu.  1  pk.  7  qt.,  and  9  bu.  3  pk.  5  qt. 


Explanation. 
The  sum  of  the  number  of  quarts  is  20 ; 
this   is   equal    to   2   pecks    and   4    quarts. 
Write  the  4  quarts  and  add  the  2  pecks  to 
the  pecks  given  in  the  second  colump. 

The  sum  of  the  number  of  pecks  is  11 ; 
this  is  equal   to   2   bushels   and  3  pecks. 
Write  the  3  pecks,  and  add  the  2  bushels  to 
the  bushels  given  in  the  third  column. 

The  sum  of  the  number  of  bushels  is  27,  which  is  wTitten  in  its 


Operation. 
7  bu.  2  pk.  5  qt. 
3  bu.  3  pk.  3  qt. 
6  bu.  1  pk.  7  qt. 
9  bu.  3  pk.  5  qt. 
27  bu.  3  pk.  4  qt. 


place. 

The  sum  is  27  bu.  3  pk.  4 

It. 

46. 

Problems. 

1.  Add. 

2.  Add. 

6  bu.  2  pk.  6  qt. 

5  bu.  0  pk.  7  qt. 

4  bu.  2  pk.  2  qt. 

4  bu.  1  pk.  6  qt. 

5  bu.  2  pk.  2  qt. 

3  bu.  0  pk.  1  qt. 

6  bu.  3  pk.  7  qt. 

5  bu.  0  pk.  5  qt. 

4  bu.  3  pk.  3  qt. 

1  bu.  1  pk.  2  qt. 

8  bu.  2  pk.  6  qt. 

2  bu.  3  pk.  3  qt. 

7  bu.  0  pk.  5  qt. 

4  bu.  2  pk.  4  qt. 

5  bu.  1  pk.  4  qt. 

2  bu.  0  pk.  6  qt. 

8  bu.  0  pk.  6  qt. 

6  bu.  0  pk.  5  qt. 

7  bu.  3  pk.  2  qt. 

4  bu.  1  pk.  0  qt. 

6  bu.  3  pk.  1  qt. 

7  bu.  0  pk.  2  qt. 

6  bu.  1  pk.  7  qt. 

3  bu.  2  pk.  2  qt. 

7  bu.  2  pk.  0  qt. 

3  bu.  1  pk.  2  qt. 

6  bu.  1  pk.  6  qt. 

3  bu.  2  pk.  1  qt. 

(a)  Find  the  sum  of  the  two  sums. 


PART    II.  167 

Algebraic  Addition. 

47.  A  coefficient  is  a  number  that  indicates  how  many 
times  a  literal  quantity*  is  to  be  taken ;  thus,  in  the  expres- 
sion 4:ab,  4  is  the  coefficient  of  ab.-f 

When  no  coefficient  is  expressed,  it  is  understood  that  1  is  the 
coefficient;  thus,  in  the  expression  4:a-\-b,  the  coefficient  of  6  is  1. 

48.  The  terms  of  an  algebraic  expression  are  the  parts 
that  are  separated  by  the  sign  +  or  -.  There  are  three 
terms  in  the  following :  a&  +  3c  +  4ahc.  There  are  only  two 
terms  in  the  following:  8a  x  46+  5a  -^  6h. 

49.  Positive  terms  are  usually  preceded  by  the  plus  sign. 
60.  Negative  terms  are  preceded  by  the  minus  sign. 

If  no  sign  is  expressed,  the  term  is  understood  to  be  positive. 

51.  When  the  literal  part  of  two  or  more  terms  is  the 
same,  the  terms  are  said  to  be  similar. 

52.  Problems. 

Unite  the  terms  in  each  of  the  following  algebraic  expressions 
into  one  equivalent  term : 

1.  bx -\- Zx -]- 2x  =  5.  4a&  +  2ab  +  3a5  = 

2.  4:X-\-5x-Sx=  6.  2ah  +  5ah  -  4:ab  = 

3.  6a  ~2a  +  4a=  7.  3bc  -  6hc  +  6hc  = 

4.  Sh  +  4:h-2h=  8.  hx  +  2hx  -f-  Sbx  = 

53.  Language  Exercise. 

Referring  to  the  problems  given  above,  use  the  following  words 
in  complete  sentences:  Coefficient,  terms,  positive,  negative,  similar, 
literal. 

*  The  word  quantity  in  algebra  means  number.  The  expression  literal  quantity 
means  number  expressed  by  letters. 

fThe  term  coefficient  is  sometimes  applied  to  the  literal  part  of  an  expression; 
thus,  in  the  expression  abc,  ab  is  the  coefficient  of  c.  Usually,  however,  the  term 
coefficient  has  reference  to  the  numerical  coefficient. 


168  COMPLETE    ARITHMETIC. 

Algebraic  Addition. 

54.  Kegarding  the  following  positive  numbers  as  represent- 
ing gains  and  the  negative  numbers  as  representing  losses, 
find  the  total  gain  (or  loss)  in  each  case ;  that  is,  find  the 
algebraic  sum  of  the  numbers  in  each  group : 

1.  2.  3.  4.  5.  6. 

70  85  45  8a  96  2ah 

25        -35       -65  3a        ~_3&      -  6«& 

95        ~50       ^ 

Note. — The  positive  sums  of  Nos.  1,  2,  4,  and  5  indicate  actual 
gain ;  the  negative  sums  of  Nos.  3  and  6  indicate  actual  loss. 

55.  Regarding  each  of  the  following  positive  numbers  as 
representing  a  rise  and  each  of  the  negative  numbers  as  rep- 
resenting a  fall  of  the  mercury  in  a  thermometer,  find  the 
total  rise  (or  fall)  in  each  case ;  that  is,  find  the'  algebraic 
sum  of  the  numbers  in  each  group : 


1. 

2. 

3. 

4. 

5. 

6. 

16 

12 

8 

8a 

ih 

5c 

10 

-4 

4 

2a 

^u 

3c 

5 

6 

-16 

3a 

4& 

-12c 

31 

14 

-4 

Note. — The  positive  sums  of  Nos.  1,  2,  4,  and  .5  indicate  actual 
rise ;  the  negative  sums  of  Nos.  3  and  6  indicate  actual  fall. 


56.  Problems. 

Find  the  sum : 

1. 

2. 

3. 

12  +  4-6 

3a  +  2&+    c 

2a6  +  Qhc 

4-h6-3 

2a-    &  +  3c 

3a6+    he 

5-2  +  8 

a  +  5&  -  6c 

-ah  +  Uc 

PART   II. 
Geometry. 


169 


57.  A  circle  is  a  plane  figure  bounded  by  a  curved  line, 
every  point  in  which  is  equally  distant  from  a  point  within, 
called  the  center. 

58.  The  line  that  bounds  a  circle  is  a  circumference. 

59.  A  straight  line  passing  through  the  center  of  a  circle 
and  ending  in  the  circumference  is  a  diameter. 

60.  A  straight  line  from  the  center  of  a  circle  to  the  cir- 
cumference is  a  radius.  ^ 

61.  Any  part  of  a  circumference  is  an  arc. 

62.  For  the  purpose  of  measurement,  every 
circumference  is  considered  as  divided  into  360 
equal  parts,  called  degrees. 

63.  Two  lines  meeting  at  a  point  form 
an  angle.  The  point  in  which  the  two 
lines  meet  is  the  vertex  of  the  angle. 

64.  Every  angle'  may  be  regarded  as 
having  its  vertex  at  the  center  of  a  circle, 
and  the  angle  is  measured  by  the  part  of 
the  arc  intercepted;    thus,  the  angle  hlk  is 
measured  by  the  arc  mn. 

65.  The  angle  efg  is  an  angle  of  90  degrees, 
called  also  a  right  angle.  The  angle  abc  is  an 
angle  of  90  degrees. 

66.  All  the  angles  about  a  point  together  equal  four  right 
angles. 


170 


COMPLETE    ARITHMETIC. 


67.  fliscellaneous  Review. 

1.  The  angle  ahd  is  an  angle  of  about 
—  degrees. 

The  angle  dhc  is  an  angle  of  —  degrees. 
The    angle    abd  +  the    angle    dhc  ■ 
angle  ahc. 

The  angle  ahc  is  an  angle  of  —  degrees. 

2.  The  angle   mln  is  an   angle    of 
about  —  degrees. 

The  angle  mln  +  the  angle  nls  =  the 
angle  mis. 

The  angle  mis  is  an  angle  of  about  — . 

3.  Copy  the  following  figures  and  add  by  column  and  by  line. 
Prove  by  comparing  the  sum  of  the  sums  of  the  columns  with  the 
sum  of  the  sums  of  the  lines.  That  pupil  who  can  solve  this  prob- 
lem without  an  error,  on  first  trial,  has  taken  an  important  step 
toward  making  himself  valuable  as  an  accountant. 


Mon. 

Tues. 

Wed. 

Thur. 

Frid. 

Sat. 

Total. 

A 

2.70 

2.95 

2.80 

3.00 

2.65 

2.45 

B 

3.43 

3.12 

3.26 

3.62 

3.28 

3.39 

C 

3.00 

2.90 

3.15 

3.20 

2.95 

3.05 

D 

2.00 

1.76 

2.22 

1.93 

1.98 

■1.87 

E 

4.15 

4.25 

4.15 

4.35 

4.45 

4.25 

F 

3.00 

3.30 

3.12 

3.18 

3.24 

3.15 

G 

5.10 

4.90 

4.95 

5.05 

5.15 

4.95 

H 

2.10 

2.12 

2.20 

2.04 

2.06 

2.25 

I 

3.50 

3.60 

3.40 

3.30 

3.50 

3.60 

K 

3.05 

2.90 

3.15 

^.95 

3.15 

2.00 

Total .  . 

1 

SUBTEACTTOK 

68.  Subtraction  (in  arithmetic)  is  the  process  of  taking 
one  number  from  (out  of)  another. 

^OTE  1. — The  word  number,  as  here  used,  stands  for  measured 
magnitude,  or  number  of  things. 

69.  The  minuend  is  the  number  from  which  another 
number  is  taken. 

70.  The  subtrahend  is  the  number  taken  from  another 
number. 

71.  The  difference  is  the  number  obtained  by  subtracting. 

72.  The  sign  — ,  which  is  read  mimts,  indicates  that  the 
number  that  follows  the  sign  is  to  be  taken  from  (out  of) 
the  number  that  precedes  it ;  thus,  8-3,  means,  that  3  is  to 
be  taken  from  (out  of)  8. 

73.  Principles. 

1.  Only  like  numhcrs  can  he  subtracted. 

2.  The  denomination  of  the  difference  is  the  same  as  that 
of  the  minuend  and  the  subtrahend. 

74.  Primary  Facts  of  Subtraction. 

There  are  eighty-one  primary  facts  of  subtraction  which 

should  be  learned  while  learning  the  facts  of  addition.    (Note  2, 

p.  443.) 

171 


172  COMPLETE    ARITHMETIC. 

75.  Examples  of  Subtraction. 


1. 

2. 

3. 

2687 

57.38 

$675.46 

1298 

28.146 

$282.75 

1389 

29.234 

$392.71 

4. 

5.. 

6. 

576  pounds 

25  bu.  3  pk.  5 

qt. 

25a+  6h 

288  pounds 

12  bu.  1  pk.  7 

qt. 

10  a  +  2  b 

288  pounds 

13  bu.  1  pk.  6 

qt. 

na-\-4b 

76.  Observe  that  in  written  problems  in  subtraction  the 
subtrahend  is  usually  placed  under  the  minuend  and  the 
difference  under  the  subtrahend;  and  that,  as  in  addition, 
the  units  of  the  same  order  are  written  in  the  same  column. 

1.  In  example  2,  what  figures  represent  units  of  the  third 
decimal  order?  of  the  second  integral  order?  of  the  first 
decimal  order  ?  of  the  first  integral  order  ? 

2.  In  example  5,  what  figures  represent  units  of  the  first 
integral  order  ?  of  the  second  integral  order  ? 

77.  Observe  that  in  subtraction  of  denominate  numbers 
the  figures  that  stand  for  units  of  the  same  denomination 
and  order  are  usually  written  in  the  same  column. 

1.  In  example  4,  what  figures  represent  tens  of  pounds  ? 
hundreds  of  pounds  ? 

2.  In  example  5,  what  figures  represent  bushels  and  units 
of  the  first  order  ? 

78.  Observe  that  in  both  addition  and  subtraction  the 
decimal  points,  if  there  are  any,  usually  appear  in  column. 


PART    II.  173 

Subtraction— Simple  Numbers. 

79.  Find  the  difference  of  8274  and  5638. 

Operation.  Explanation. 

8274  Eight  is  greater  than  4.     In  the  minuend,  take  one 

5638      unit  of  the  second  order  from  the  7  units  of  the  second 

()c.oa      order.      This  unit  of  the  second  order,  combined  with 

the  4  units  of  the  first,  makes  14  units  of  the  first  order. 

Eight  units  of  the  first  order  from  14  units  of  the  first  order  leave  6 

units  of  the  first  order.     Three  units  of  the  second  order  from  6 

(7—1)  units  of  the  second  order  leave  3  units  of  the  second  order. 

Six  is  greater  than  2.     In  the  minuend  take  one  unit  of  the  fourth 

order  from  the  8  units  of  the  fourth  order.     This  unit  of  the  fourth 

order,  combined  with  the  2  units  of  the  third  order,  makes  12  units 

of  the  third  order.     Six  units  of  the  third  order  from  12  units  of 

the  third  order  leave  6  units  of  the  third  order.     Five  units  of  the 

fourth  order  from  7  (8  —  1)  units  of  the  fourth  order  leave  2  units 

of  the  fourth  order. 

The  difference  of  8274  and  5638  is  2636. 

80.  Problems. 

1.  From  35642  subtract  12456. 

2.  From  875^44  subtract  64358. 

3.  From  90070  subtract  13256. 

4.  From  8164  subtract  3275. 

5.  From  the  sum  of  8539,  2647,  3984,  1461,  7353,  6016, 
and  2364,  subtract  22364. 

6.  From  the  sum  of  1352,  3425,  2640,  3724,  6575,  7360, 
and  6276,  subtract  21352. 

7.  From  6  thousand  7  hundred  25,  subtract  1  thousand 
8  hundred  36. 

8.  From  seven  thousand  four  hundred  sixty-five,  subtract 
two  thousand  three  hundred  fifty-four. 

(a)  Find  the  sum  of  the  eight  differences. 


174  ,      COMPLETE    ARITHMETIC. 

Subtraction— Decimals. 

81.  Find  the  difference  of  28.36  and  15.432. 

Operation.  "     Explanation. 

28.36  One  unit  of  the  second  decimal  order  (1  from  6) 

15.432       equals  10  units  of  the  third  decimal  order.     Two  from 

12  9*^8      ^^  leaves  8. 

Three  from  5  (6—1)  leaves  2. 

Four  is  greater  than  3.  One  unit  of  the  first  integral  order 
(1  from  8)  equals  10  units  of  the  first  decimal  order.  10  +  3  m  13. 
Four  from  13  leaves  9. 

Five  from  7  (8  —  1)  leaves  2.     One  from  2  leaves  1. 

The  difference  of  28.36  and  15.432  is  12.928. 

82.  Problems. 

1.  From  100  take  .3456.  6.  100  -  44.764. 

2.  From  100  take  5.246.  7.  250  -  159.63. 

3.  From  100  take  44.236.  8.  250  -  36.75. 

4.  From  100  take  .6544.  9.  250  -  140.37. 

5.  From  100  take  4.754.  10.  250  -  163.25. 

(a)  Find  the  sum  of  the  ten  differences. 

83.  Miscellaneous. 

1.  The  sum  of  two  numbers  is  3.7464;  one  of  the  num- 
bers is  1.3521.     What  is  the  other  number? 

2.  The  difference  of  two  numbers  is   2.3254;  the  less 
number  is  7.6746.     Wliat  is  the  greater  number? 

3.  The  difference  of  two  numbers  is  2.3943;  the  greater 
number  is  10.     What  is  the  less  number? 

4.  From  10  subtract  7.6744. 

(b)  Find  the  sum  of  the  four  results. 

To  THE  Pupil. — Work  with  care.  Make  no  errors.  The  sum  of 
the  results  should  be  correct  on  first  trial. 


PART   II. 


175 


Subtraction— United  States  Money. 

84.  Find  the  difference  of  $27.25  and  $14.51. 
Explanation. 
One  cent  from  5  cents  =  4  cents. 
Five  dimes  from  12  dimes  =  7  dimes. 
Four  dollars  from  6  dollars  (7  — 1)  =  2  dollars. 
One  ten-dollars  from  2  ten-dollars  =  1  ten-dollars. 
The  difference  of  $27.25  and  .*$14.51  is  $12.74. 


Operation. 

$27.25 
$14.51 

$12.74 


85.  Problems — Addition  and  Subtraction.* 


Find  the  sum  that  each  depositor  has  to  his  credit : 
A.  B. 

Deposit 
Check     $30.50 


Deposit 

$254.20 

Deposit 

38.60 

Check    $12.50 

Check      10.80 

Check        3.60 

Check         5.40 

Balance  

. 

C. 

Deposit 

$745.80 

Check    $87.50 

Check      89.20 

Check      96.40 

Check      94.60 

Deposit 

45.80 

Balance  

Check 

Deposit 

Check 

Check 

Balance 


21.75 

18.34 
6.24 

D. 


$175.30 


54.20 


E.  Deposit,  $1000. 

F.  Deposit,  $864. 


Deposit  $824.70 

Check    $69.50 

Check      78.25 

Check      81.66 

Deposit 

Check      93.76 

Balance  

Checks,  $275,  $324,  $400. 
Checks,  $375,  $146,  $279. 


61.40 


(a)  Find  the  amount  of  the  six  balances. 


*  That  bank  clerk  who  makes  one  error  a  day  in  carrying  out  his  balances, 
which  he  does  not  himself  discover  and  correct,  will  not  retain  his  position. 


176  COMPLETE    ARITHMETIC. 

Subtraction— Denominate  Numbers. 

86.  Find  the  difference  of  15  yd.  2  ft.  4  in.  and  8  yd.  1  ft. 
10  in. 

Operation.  Explanation. 

15  yd.  2  ft.     4  in.  Ten  inches  are  more  than  4  in.;    1  ft. 

8  yd.  1  ft.  10  in.      (from  the  2  ft.)  equals  12  in.;   12  in.  and 

7  vd   0  ft      6  in       ^  ^^^'  ^^'®  ^^  "^•'  ^^  ^^'  ^^^^  1^  i^*  leave  6 
in. 
One  ft.  from  1  ft.  (2-1)  leaves  0  ft. 
Eight  yd.  from  15  yd.  leave  7  yd. 

The  difference  of  15  yd.  2-  ft.  4  in.  and  8  yd.  1  ft.  10  in.  is  7  yd. 
6  in. 

87.  Problems. 

1.  From  12  yd.  1  ft.  8  in.  subtract  5  yd.  2  ft.  3  in. 

2.  From  8  yd.  2  ft.  6  in.  subtract  5  yd.  1  ft.  10  in. 

3.  From  9  yd.  1  ft.  subtract  2  yd.  2  ft.  7  in. 

4.  From  6  yd.  2  ft.  5  in.  subtract  4  yd.  8  in. 

5.  From  7  yd.  subtract  5  yd.  1  ft.  1  in. 

6.  From  7  yd.  1  ft.  4  in.  subtract  4  yd.  9  in. 

7.  From  11  yd.  6  in.  subtract  4  yd.  1  ft.  2  in. 

8.  From  10  yd.  2  ft.  subtract  7  yd.  5  in. 

(a)  Find  the  sum  of  the  eight  differences. 

88.  Problems. 

1.  How  many  days  from  April  25  to  May  1  ? 

2.  How  many  days  from  April  25  to  May  10  ? 

3.  How  many  days  from  April  25  to  May  20  ? 

4.  How  many  days  from  April  25  to  June  1  ? 

5.  How  many  days  from  April  25  to  June  10  ? 

6.  How  many  days  from  April  25  to  June  30  ? 

7.  How  many  days  from  April  25  to  July  5  ? 

(b)  Find  the  sum  of  the  seven  answers. 


Tues. 

Wed. 

Thurs. 

Fri. 

Sat. 

60 

20 

la 

-4& 

-12c 

-20 

80 

50 
-30 

2a 

5  a 

-6& 
2h 

-  6c 

-  6c 

PART   II.  177 

Algebraic  Subtraction. 

89.  Eegarding  the  following  minuends  as  representing  A's 
gain  (or  loss),  and  the  subtrahends  as  representing  B's  gain 
(or  loss),  subtract  B's  from  A's.* 

Mon. 

A,  80 

B,  30 

50 

Note. — The  positive  differences  for  Monday,  Tuesday,  Thursday, 
and  Friday  indicate  that  A's  gain  was  greater  (or  his  loss  less)  than 
B's.  The  negative  differences  for  Wednesday  and  Saturday  indicate 
that  A's  gain  was  less  (or  his  loss  greater)  than  B's. 

90.  Eegard  the  following  minuends  as  representing  dis- 
tances one  boat  sails  from  a  given  point,  and  the  subtrahends 
as  representing  distances  another  boat  sails  from  the  same 
point.  Distances  sailed  north  are  here  represented  by  posi- 
tive numbers,  and  distances  sailed  south  by  negative  num- 
bers.    Find  how  far  the  first  boat  is  from  the  second. 


1. 

2. 

3. 

4. 

5. 

6. 

1st  B., 

75 

85 

15 

8a 

-U 

-15c 

2nd  B., 

25 

-35 

35 

2a 

-9b 

-    5c 

50 

120 

-20 

Note. — The  positive  differences  in  Nos.  1,  2,  4,  and  5  indicate 
that  the  first  boat  is  north  of  the  second  boat.  The  negative  differ- 
ences in  Nos.  3  and  6  indicate  that  the  first  boat  is  south  of  the 
second  boat. 

91.  Eule  for  algebraic  subtraction :  Conceive  the  sign  (or 
signs)  of  the  subtrahend  to  be  changed  (—  to  +  and  +  to  — ), 
then  proceed  as  in  addition. 

*Seepagel68,  Art.  54. 


178  COMPLETE    ARITHMETIC. 

Algebraic  Subtraction. 

1.  A  gained  $1200  and  lost  $250  ;  B  gained  $500  and  lost 
$350.  How  much  more  was  A's  wealth  increased  by  the 
two  transactions  than  B's  ? 

$1200 -$250  12^-56  Ua-Sh 

$500  -  $350  5a-7b  6a-5b 

$700  +  $100  ^  ~ 

2.  C  gained  $1500  and  $650;  D  gained  $600  and  lost 
$250.  How  much  more  was  C's  wealth  increased  by  the 
two  transactions  than  D's  ? 

$1500 +  $650         lb  a  +  13b  IS  a -\- 6  b 

$600 -$250  6a-    5b  4.a-Sb 


+ 

3.  E  gained  $1300  and  lost  $450;  F  gained  $400  and 
$250.  How  much  more  was  E's  wealth  increased  by  the 
two  transactions  than  F's  ? 

$1300 -$450  IS  a -9b  17  a -8b 

$400 +  $250  4:a  +  5b  4a  +  6& 

$900  -  $700 

4.  G  gained  $1200  and  lost  $500  ;  H  gained  $900  and 
lost  $100.  How  much  more  was  G's  wealth  increased  by 
the  two  transactions  than  H's  ? 

$1200 -$500  12a -5b  16a-Sb 

$900  -  $100  9a-     b  4^-25 

$300  -  $400 

5.  Eeview  the  foregoing  and  observe  that  in  every  instance 
subtracting  a  positive  number  is  equivalent  to  adding  an 
equal  negative  number,  and  subtracting  a  negative  number 
is  equivalent  to  adding  an  equal  positive  number. 


B 


PART    II.  179 

Geometry. 

92.  Direction  of  Lines. 

A 

M N 

O P 

D 


C 

1.  When  one  straight  line  meets  another  straight  hne  in 
such  a  manner  that  two  right  angles  are  formed  by  the  lines, 
the  two  lines  are  said  to  be  perpendicular  to  each  other. 

2.  Two  lines  side  by  side  extending  in  the  same  direction 
are  said  to  be  parallel. 

3.  Of  the  lines  given  above : 

AC  is  to  BD. 

BD  is to  AC. 

MN  is to  OP. 

4.  A  line  extending  in  the  direction  of  the  horizon  is  said 
to  be  horizonidX.  A  line  on  the  floor  of  the  room  is  hori- 
zontal; a  line  on  the  ceiling  is  horizontal;  a  line  on  the 
blackboard,  every  point  in  which  is  equally  distant  from  the 
floor,  is  horizontal.  For  convenience,  lines  drawn  upon 
paper,  that  are  parallel  with  the  top  and  bottom  of  the  paper, 
may  be  regarded  as  representing  horizontal  lines. 

5.  A  line  suspending  a  piece  of  lead  (plumbum)  is  called 
a  plumb-line.  A  line  in  the  direction  of  a  plumb-line  is  said 
to  be  vertical.  A  vertical  line  is  perpendicular  to  a  hori- 
zontal line.  Lines  on  the  blackboard  may  or  may  not  be 
vertical  or  horizontal.  For  convenience,  lines  drawn  upon 
paper  that  are  parallel  with  the  sides  of  the  paper  may  be 
regarded  as  vertical. 


180 


COMPLETE    ARITHMETIC. 


93.  Miscellaneous  Reviews. 

1.  An  angle  that  is  equal  to  one  half  of  a 
right  angle,  is  an  angle  of  degrees. 

2.  The  angle  ADB  is  an  angle  of  de- 
grees. 

3.  If  from  a  right  angle,  an   angle   of   30 
remaining  angle  is  an 


D 


degrees  be 
angle  of  — 


taken,  the 
—  degrees. 


4.  The  angle  FHG  is  an  angle  of 

grees. 

5.  During  the  month  of  November,  1897,  there  were  con- 
sumed at  the  Illinois  Institution  for  the  Education  of  the 
Blind,  64  loads  of  coal.  The  weight  of  each  load  in  pounds 
is  given  below.  Find  the  total  weight. 

6100  8020               5490  5190 

8380  6860               6800  7130 

4850  6230               6560  7090 

8010  6780               6690  7790 

7080  6980              5780  6810 

6620  6240               6980  8600 

6450  6420              5990  9100 

6570  6310              4740  6740 

7950  6300              5520  5380 

4750  6530               3630  7640 

8840  6950              4930  5650 

7290  6980              5150  5900 

4960  4920               6420  6200 

8330  5880              6770  6620 

6300  7030               6220  7170 

7080  5160              6020  9210 


MULTIPLiCATIOK 

94.  Multiplication  is  the  process  of  taking  a  number  (of 
things)  a  number  of  times. 

The  word  number  as  first  used  in  the  above  statement  stands  for 
measured  magnitude.  The  second  word  nuinher  does  not  stand  for 
measured  magnitude,  but  rather  for  pure  number,  representing 
simply  the  times  the  number  (of  things)  is  to  be  repeated. 

95.  The  multiplicand  is  thie  number  (of  things)  taken  or 
repeated.* 

96.  The  multiplier  is  the  number  that  shows  how  many 
times  the  multiplicand  is  to  be  repeated. 

97.  The  product  is  the  number  (of  things)  obtained  by 
multiplying. 

98.  The  sign,  x,  which  is  read  multiplied  hi/,  indicates 
that  the  number  preceding  the  sign  is  a  multiplicand,  and 
the  number  following  it  a  multiplier. 

For  other  uses  of  this  sign,  see  notes  3,  4,  and  5,  page  443. 

99.  Principles. 

1.  The  multiplier  is  always  an  abstract  number. 

2.  The  denomination  of  the  product  is  always  the  same 
as  that  of  the  multiplicand. 

100.  Primary  Facts  of  Multiplication. 

There  are  sixty-four  primary  facts  of  multiplication.  See 
note  6,  page  444. 

*  "  The  multiplicand,  however  written,  must  always  be  understood  to  express 
measured  quantity;  it  is  always  concrete."— Psychology  of  Number,  McCleUan  &  Deivey, 
page  76. 

181 


182  COMPLETE    ARITHMETIC. 

101.  Examples  of  Multiplication. 

1.  2.                              3. 

4865  37.258                   $375.42 

3  3                               3 


14595  111.774  $1126.26 

4.  5.  6. 

364  tons  61  yd.  2  ft.  5  in.  8a  -    6b 
5                                       5  5 


1820  tons       309  yd.  0  ft.  1  in.         4:0  a  -  30  b 

102.  Observe  that  in  each  of  the  above  examples  the 
multiplier  is  a  pure  number. 

103.  Observe  that  in  each  of  the  above  examples  the 
denomination  of  the  product  is  the  same  as  the  denomina- 
tion of  the  multiplicand. 

104.  Multiplication  and  Addition  Compared. 

Find  the  sum  of  each  of  the  following  groups  of  numbers  and 
compare  the  result  with  the  product  in  the  corresponding  problem 
in  article  101  : 

1.  2.  3. 


4865 

37.258 

$375.42 

4865 

37.258 

$375.42 

4865 

37.258 

$375.42 

4. 

5. 

6. 

364  tons 

61 

yd.  2  ft.  5 

in. 

8a-  6b 

364  tons 

61 

yd.  2  ft.  5 

in. 

8a  -  6b 

364  tons 

61 

yd.  2  ft.  5 

in. 

8a  -  6b 

364  tons 

61 

yd.  2  ft.  5 

in. 

8a  -  6b 

364  tons 

61 

yd.  2  ft.  5 

in. 

8a  -  6b 

PART    II.  183 

Multiplication— Simple  Numbers. 

105.  Find  the  product  of  563  and  7. 
Operation.  Explanation. 

563  Seven  times  3  units  of  the  first  order  are  21  units  of  the 

7       first  order ;  they  are  equal  to  1  unit  of  the  first  order  and 

oQ^-|       2  units  of  the  second  order.     Write  the  1  unit  of  the  first 

order  and  add  the  2  units  of  the  second  order  to  the  next 

partial  product. 

Seven  times  6  units  of  the  second  order  are  42  units  of  the  second 
order ;  42  +  2  =  44  ;  44  units  of  the  second  order  equal  4  units  of 
the  second  order  and  4  units  of  the  third  order.  Write  the  4  units 
of  the  second  order  and  add  the  4  units  of  the  third  order  to  the 
next  partial  product. 

Seven  times  5  units  of  the  third  order  are  35  units  of  the  third 
order;  35  +  4  =  39 ;  39  units  of  the  third  order  equal  9  units  of  the 
third  order  and  three  units  of  the  fourth  order.  Write  the  9  units 
of  the  third  order  and  the  3  units  of  the  fourth  order. 

The  product  of  563  and  7  is  3941. 

106.  Find  the  product  of  3426  and  57. 
Operation.  Explanation. 

3426  Seven  times  3426  equals  23982.     Fifty  times  3426 

57       equals  171300.     Fifty  times  the  number  plus  7  times 
nn(\c,.')       the  number  equals  57  times  the  number.     Therefore 
71  ^0         ^<idi"g  23982  and  171300  gives  57  times  3426. 
The  product  of  3426  and  57  is  195282. 
195282 

107.  Problems. 

1.  Multiply  3241  by  27.  6.  6521  x  54  = 

2.  Multiply  6759  by  27.  7.  3572  x  74  - 

3.  Multiply  4328  by  36.  8.  6428  x  74  = 
4  Multiply  5672  by  36.  9.  3521  x  29  = 
5.  Multiply  3479  by  54.  10.  6479  x  29  = 

(a)  Find  the  sum  of  the  ten  products. 


184  COMPLETE    ARITHMETIC. 

Multiplication — Decimals. 

108.  Find  the  product  of  728.37  and  .6. 
Operation.  Explanation. 

72^8.37  To  multiply  by  .6  means  to  take  6  times  1  tenth 

.6      of  the  number.     One  tenth  of  728.37  is  72.837.     Six 
.0-7  029      ti^es  72.837  equals  437.022. 

Note  1.— The  separatrix  is  used  to  indicate  the 
place  of  the  decimal  point  in  the  number  that  is  one  tenth  of  the 
multiplicand. 

Note  2. — The  decimal  point  should  be  written  in  the  product 
when,  in  the  process  of  multiplication,  the  place  is  reached  where  it 
belongs.  Do  not  multiply  all  the  figures  and  then  attempt  to  deter- 
mine the  place  of  the  point. 

109.  Find  the  product  of  746.2  and  .25. 

Operation.  Explanation. 

7^46.2  To  multiply  by  .25  means  to  take  25  times  1  hun- 

.25  dredth  of  the  number. 

37  32Q  ■       One  hundredth  of  746.2  is  7.462.    Five  times  7.462 

149  24  equals  37.310.     Twenty  times  7.462  equals  149.24. 

37.310  +  149.24  =  186.550. 

186.550 

110.  Observe  that  when  a  multiplication  of  decimals  is 
complete,  the  number  of  decimal  places  in  the  product  is 
equal  to  the  number  of  decimal  places  in  the  multiplicand 
and  multiplier. 

111.  Problems. 

1.  Multiply  324.6  by  .7.  6.  324.6  x  54  = 

2.  Multiply  324.6  by  .27.  7.  324.6  x  .48  = 

3.  Multiply  324.6  by  2.7.  8.  324.6  x  6.3  = 

4.  Multiply  324.6  by  27.  9.  324.6  x  3.07  = 

5.  Multiply  324.6  by  5.4.         10.  324.6  x  .08  = 
(a)  Find  the  sum  of  the  ten  products. 


PART    II. 


185 


Multiplication— United  States  Money. 
112.  Find  the  product  of  $34.50  and  53.4. 


Operation. 

$3^4.50 
53.4 
$13,800 
$103.50 

$1725.0 
$1842.300 


Explanation. 
To  multiply  by  53.4,  means  to  take  53  times  the 
multiplicand  plus  4  tenths  of  the  multiplicand. 
One  tenth  of  ^34.50  is  ^3.45. 
Four  tenths  of  $34.50  is  113.80. 
Three  times  134.50  is  $103.50. 
Fifty  times  $34.50  is  $1725. 
The  sum  of  the  partial  products  is  $1842.30. 

Practical  application  of  the  foregoing. 

If  one  acre  of  land  costs  $34.50,  how  much  will  53.4  acres 

cost  ? 
One  tenth  of  an  acre  costs  $3.45. 
Four  tenths  of  an  acre  cost  $13.80. 
Three  acres  cost  $103.50. 
Fifty  acres  cost  $1725. 
53.4  acres  cost  $1842.30. 

113.  Complete  the  following  bill  and  find  the  amount: 

Inst,  for  the  Blind, 
1897.  To  Geo.  E.  Sybrant,  Dr. 


Dec. 

4 

27  bbl.  Apples       @  $2.25 

I 

10 

56  bush.  Potatoes  @       .52 

( 

12 

13  bush.  Beans      @     1.75 

< 

16 
18 

34  bush.  Turnips  @       .35 
50  bbl.  Flour         @     4.90 

( 

21 

74  lb.  Butter         @       .19 

' 

22 

53  lb.  Tea              @       .42 

i 

24 

37  bush.  Onions    @       .55 

u 

31 

58  lb.  Ham            @       .14 

To  THE  Pupil. — Remember  that  any  inaccuracy  in  solving  busi- 
ness problems  makes  the  work  valueless.  Accuracy  ranks  next  in 
importance  to  integrity  in  the  selection  of  an  accountant. 


186  COMPLETE    ARITHMETIC. 

Multiplication— Denominate  Numbers. 
114.  Find  the  product  of  3  tons  850  lb.  and  8. 
Operation.  Explanation. 

3  tons  850  lb.  Eight  times  850  lb.  equals  6800  lb. 

8  6800  lb.  equals  3  tons  800  lb. 


27  tons  800  lb  Write  the  800  lb.  and  add  the  3  tons  to  the 

next  partial  product. 
Eight  times  3  tons  equals  24  tons;  24  tons  plus  3  tons  equals  27 
tons.     3  tons  800  lb.  multiplied  by  8  equals  27  tons  800  lb. 

115.  Problems. 

1.  If  one  side  of  a  square  garden  measures  6  rd.  8  ft, 
what  is  the  perimeter  of  the  garden  ? 

2.  The  circumference  of  a  certain  bicycle  track  is  13,rd. 
1 2  ft.  How  far  does  the  rider  travel  who  goes  around  it  .12 
times  ? 

3.  The  length  of  a  rectangular  field  is  15  rd.  10  ft.  and 
the  width  9  rd.  8  ft.     What  is  the  perimeter  of  the  field  ? 

4.  There  is  a  walk  5  feet  wide  around  a  rectangular  grass 
plat  3  rd.  6  ft.  by  2  rd.  10  ft.  What  is  the  outside  perim- 
eter of  the  walk  ? 

5.  How  far  does  the  person  travel  who  walks  once  around 
the  grass  plat  described  in  problem  4,  if  he  keeps  his  track 
in  the  center  of  the  walk  ? 

(a)  Find  the  sum  of  the  five  answers. 

116.  Problems. 

1.  If  a  train  moves  at  the  rate  of  a  mile  in  1  min.  25  sec, 
in  how  long  a  time  will  it  move  325  miles  ? 

2.  If  the  circumference  of  a  wagon  wheel  is  15  feet  6 
inches,  how  far  will  the  wagon  move  while  the  wheel  re- 
volves 1000  times? 


PART    II.  187 

Algebraic  Multiplication. 

117.  Examples. 
No.  1.  No.  2. 

8  +  4-3  a  +  Sb-4:C 


32  +  16  -  12  4a +  12?)-  16c 

No.  3.  No.  4. 

3ft  -2b-^c  2ft  +  3&  -  5c 

f^  2(^ 


3ac^  -  2bd  +  cc/  4a£^  +  Qhd  -  lOcd 

1.  Observe  that  in  the  above  examples  we  multiply  each 
term  of  the  multiplicand  by  the  multiplier. 

2.  Prove  example  No.  1  by  uniting  the  terms  of  the  mul- 
tiplicand and  comparing  4  times  the  number  thus  obtained 
with  the  number  obtained  by  uniting  the  terms  of  the 
product. 

3.  Verify  example  No.  2  by  letting  a  =  b,b  =  ?>,  and  c  =  2. 

4.  Verify  example  No.  3  by  giving  the  following  values  to 
the  letters :  «,  =  7,  5  =  4,  c  =  3,  ^  =  5. 

5.  Verify  example  No.  4  by  giving  any  values  you  may 
choose  to  each  letter. 

118.  Problems. 

1.  Multiply  3a5  -  2bc  +  5c  by  2d. 

2.  Multiply  2ax  +  4:bx  -  ?/  by  5. 

3.  Multiply  Zbc  +  ab  -be  by  Sd. 

4.  Multiply  X  —  1/  -\-zhj  Sab. 

6.  Multiply  ax-}-bx  —  ex  by  2'i/.- 

6.  Verify  each  of  the  above  problems  by  giving  the  fol- 
lowing values  to  the  letters :  a  =  S,  b  =  2,  c  =  4,  d  =  5,  x=7, 
2/  =  6,  0  =  8. 


188  COMPLETE    ARITHMETIC. 

Algebraic  Multiplication. 

119.  Exponent. 

1.  a  X  d,  or  aa  which  means  a  multipKed  by  a,  is  usually 
written  a\     This  is  read  a  square  or  a  second  power. 

2.  b^  (to  be  read  h  cube  or  h  third  power)  means  h  taken 
three  times  as  a  factor.     It  is  h  xh  xb- 

3.  a*  (to  be  read  a  fourth  power,  or  simply  a  fourth) 
means  that  a  is  taken  four  times  as  a  factor.  It  is  a  x  cc  x 
a  X  cc- 

4.  The  small  figure  at  the  right  of  a  letter  tells  the  number 
of  times  the  letter  is  to  be  used  as  a  factor.  The  figure  so 
used  is  called  an  exponent.  When  the  exponent  is  1,  it  is 
not  usually  expressed ;  thus,  a  means  a\ 

120.  Problems. 

On  the  supposition  that  a  =  2,  b  =  3,  and  c  =  4,  find  the  numerical 
value  of  each  of  the  following  expressions : 

1.  a'-{-2ah-th'  6.  5a'h-2hc 

2.  Sa¥-}-5hc'  7.  a'b' -  c' 

3.  ^a'h'  +  ShV  •  8.  a'h'  +  c' 

4.  2a'h  f  2ay'  9.  d'b'c'  -  ab\ 

5.  3&V4-5a6  10.  2aTc'^ 

(a)  Find  the  sum  of  the  numerical  values  of  the  above. 


121. 

No.  1. 
4:ax -Y  2by  -\- c 
a' 

Examples. 

No.  2. 

Wx  -\-2by-c 
2b' 

4a'x  +  2a'by  +  d'c  6b'x  +  4%  -  2&'c 

Verify  each  of  the  above  examples  by  letting  a  =  2,b  =  S, 
c  =  4:,  X-  5,  y  =  6. 


PART    II. 

Geometry. 
122.  Parallelograms. 


189 


1.  Any  side  of  any  one  of  the  above  figures  is  parallel  to 
the  opposite  side  of  the  same  figure.  Hence  the  figures  are 
called  parallelograms. 

2.  Each  of  the  above  figures  has  four  sides.  Hence  the 
figures  are  called  quadrilaterals. 

3.  If  all  the  sides  of  a  figure  are  equal,  the  figure  is  said 
to  be  eqioilateral. 

4.  If  all  the  angles  of  a  parallelogram  are  right  angles 
(angles  of  90°)  the  figure  is  said  to  be  rectangular. 

5.  Wliich  of  the  above  figures  are  equilateral  ? 

6.  Which  of  the  above  figures  are  rectangular  ? 

7.  Which  of  the  above  figures  are  not  equilateral  ? 

8.  Which  of  the  above  figures  are  not  rectangular  ? 

9.  Which  of  the  above  figures  are  parallelograms  ? 

10.  Which  of  the  above  figures  are  quadrilaterals  ? 

11.  Can  you  draw  a  quadrilateral  that  is  not  a  parallelo- 
gram ? 

12.  Is  any  one  of  the  above  figures  an  equilateral  rect- 
angular parallelogram  ? 

13.  In  a  rhomboid  or  rhombus  two  of  the  angles  are  less 
than  right  angles  and  two  of  them  are  greater  than  right 
angles.  Convince  yourself  by  cutting  a  rhomboid  from 
paper  and  comparing  it  with  rectangular  figures  that  two  of 
the  angles  of  a  rhomboid  are  as  much  less  than  two  right 
angles  as  the  other  two  are  greater  than  two  right  angles. 


190  COMPLETE    ARITHMETIC. 

123.  Miscellaneous  Reviews. 

1.  If  one  of  the  angles  of  a  rhombus  is  an  angle  of  80 
degrees,  what  is  the  number  of  degrees  in  each  of  the  other 
angles  ? 

2.  Draw  a  rhomboid  one  of  whose  angles  is  an  angle  of 
70 ;  give  the  number  of  degrees  in  each  of  the  other  angles. 

3.  An  oblong  has  four  right  angles.  The  angles  of  a 
rhomboid  are  together  equal  to  how  many  right  angles  ? 

4.  If  an  oblong  is  a  feet  long  and  l  feet  wide,  the  number 
of  square  feet  in  the  area  is  a&.*  If  the  side  of  a  square  is 
a  feet,  the  number  of  square  feet  in  its  area  is  . 

5.  If  a  rectangular  solid  is  a  feet  long,  h  feet  wide,  and  c 
feet  thick,  the  number  of  cubic  feet  in  its  solid  contents  is 
abc.  If  the  side  of  a  cube  is  a  feet,  the  number  of  cubic 
feet  in  its  solid  contents  is  . 

6.  If  a  man  earns  h  dollars  each  week  and  spends  c  dollars, 

in  one  week  he  will  save  dollars ;  in  7  weeks  he 

will  save  dollars. 

7.  A  framed  picture,  on  the  inside  of  the  frame,  is  18  in. 
by  22  in. ;  the  frame  is  4  inches  wide.  How  many  inches 
in  the  outside  perimeter  of  the  frame  ? 

8.  Think  of  two  fields:  one  is  9  rd.  by  16  rd. ;  the  other 
is  12  rd.  by  12  rd.  How  do  the  square  rods  of  the  two 
fields  compare  ?  How  much  more  fence  would  be  required 
to  enclose  one  field  than  the  other  ? 

*  This  means  the  product  of  a  and  6.  Observe  that  it  is  the  numljer  a  (not  a  feet) 
that  we  multiply  by  the  number  b  (not  6  feet).  While  it  is  probably  true  (see  foot- 
note, p.  181)  that  the  multiplicand  always  expresses  measured  quantity,  it  is  also 
true  that  we  often  find  the  product  of  two  factors  mechanically.  Indeed  this  is 
what  we  usually  do  in  all  multiplication  of  abstract  numbers.  In  this  case  we  find 
the  product  of  a  and  6  and  know  from  former  observations  that  this  number  equals 
the  number  of  square  feet  iu  the  oblong. 


DIVISION 

124.  Division  is  (1)  the  process  of  finding  how  many 
times  one  number  is  contained  in  another  number;  or  (2), 
it  is  finding  one  of  the  equal  parts  of  a  number. 

Note.— The  word  number  as  used  above  stands  for  measured 
magnitude. 

125.  The  dividend  is  the  number  (of  things)  to  be  divided. 

Note.— Since  in  multipHcation  the  multipUcand  and  product  must 
always  be  considered  concrete  (see  foot-note,  p.  181),  then  in  division, 
the  dividend,  and  either  the  divisor  or  the  quotient,  must  be  so 
regarded. 

126.  The  divisor  is  the  number  by  which  we  divide. 

Note. — The  word  number  as  used  in  Art.  126  may  stand  for  meas- 
ured magnitude  or  for  pure  number,  according  to  the  aspect  of  the 
division  problem.  In  the  problem  324  h-  6,  if  we  desire  to  find  how 
many  times  6  is  contained  in  324,  the  6  stands  for  measured  magni- 
tude— a  number  of  things.  But  if  we  desire  to  find  one  sixth  of  324, 
then  the  6  is  pure  number,  and  is  the  ratio  of  the  dividend  to  the 
required  quotient. 

127.  The  quotient  is  the  number  obtained  by  dividing. 

Note. — If  the  divisor  is  pure  number,  the  quotient  represents 
measured  magnitude.  If  the  divisor  represents  measured  magnitude, 
the  quotient  is  pure  number. 

128.  The  sign  -^,  which  is  read  divided  hy,  indicates  that 
the  number  before  the  sign  is  a  dividend  and  the  number 
following  the  sign  a  divisor.     See  notes  7  and  8,  page  445. 

191 


192  COMPLETE    ARITHMETIC. 

129.  Examples  in  Division. 


No.  1. 

No.  2. 

$5)$1565 

5)$1665 

313 

$313 

No.  3. 

No.  4. 

2  bush.)246  bush. 

2)246  bush. 

.      123 

123  bush. 

No.  5. 

No.  6. 

2a)6ab -\- 8ac  -  12a 

2)6ah-{-8ac-12a 

36    +4c    -    6  3ab-}-4ac-    6a 


1.  In  example  No.  1,  we  are  required  to  find 

2.  In  example  No.  2,  we  are  required  to  find 

3.  In  example  No.  3,  we  are  required  to  find 

4.  In  example  No.  4,  we  are  required  to  find 

5.  In  example  No.  5,  we  are  required  to  find 

6.  In  example  No.  6,  we  are  required  to  find 


Note.— Let  it  be  observed  that  all  the  examples  given  on  this  page,  indeed  all 
division  problems,  may  be  regarded  as  requirements  to  find  how  many  times  one 
number  of  things  is  contained  in  another  number  of  like  things.  Referring  to 
example  No.  2  given  above :  If  one  were  required  to  find  one  fifth  of  1565  silver 
dollars,  he  might  first  take  5  dollars  from  the  1565  dollars,  and  put  one  of  the  dol- 
lars taken  in  each  of  five  places.  He  might  then  take  another  five  dollars  from  the 
number  of  dollars  to  be  divided,  and  put  one  dollar  with  each  of  the  dollars  first 
taken.  In  this  manner  he  would  continue  to  distribute  fives  of  dollars  until  all 
the  dollars  had  been  placed  in  the  five  piles.  He  would  then  count  the  dollars  in 
each  pile.  Observe,  then,  that  one  fifth  of  1565  dollars  is  as  many  dollars  as  $5  is 
contained  times  in  $1565.  It  is  contained  313  times ;  hence  one  fifth  of  1565  dollars 
is  313  dollars. 

It  is  not  deemed  advisable  to  attempt  such  an  explanation  as  the  foregoing  with 
young  pupils ;  but  the  more  mature  and  thoughtful  pupils  may  now  learn  that  it  is 
possible  to  solve  all  division  problems  by  one  thought  process— finding  how  many 
times  one  nmnber  of  things  is  contained  in  another  number  of  like  things. 

*Fill  the  blank  with  the  words,  how  many  times  five  dollars  are  contained  in  $1565. 
tFill  the  blank  with  the  words,  one  fifth  of  $1665. 


PART    II.  193 

Division— Simple  Numbers. 

130.  Find  the  quotient  of  576  divided  by  4. 

''  Short  Division."  Explanation  No.  1. 

4)576  ^^®  fourth  of  5  hundred  is  1  hundred  with  a  remain- 

der  of  1  hundred ;  1  hundred  equals  10  tens ;  10  tens 

plus  7  tens  are  17  tens.  One  fourth  of  17  tens  is  4  tens 
with  a  remainder  of  1  ten  ;  1  ten  equals  10  units ;  10  units  plus  6  units 
are  16  units.  One  fourth  of  16  units  is  4  units.  Hence  one  fourth 
of  576  is  144. 

Explanation  No.  2. 
Four  is  contained  in  5  hundred,  1  hundred  times,  with  a  remainder 
of  1  hundred;  1  hundred  equals  10  tens;  10  tens  and  7  tens  are  17 
tens.  Four  is  contained  in  17  tens,  4  tens  (40)  times  with  a  remain- 
der of  1  ten;  1  ten  equals  10  units;  10  units  and  6  units  are  16 
units.  Four  is  contained  in  16  units  4  times. 
Hence  4  is  contained  in  576,  144  times. 

131.  Find  the  quotient  of  8675  divided  by  25. 
"Long  Division."  Explanation. 

25')8675('347  Twenty-five  is  contained  in  86  hundred,  3 

yg  hundred  times  with  a  remainder  of  11  hundred; 

11   hundred  equal  110  tens;  110  tens  plus  7 
tens  equal  117  tens.     Twenty-five  is  contained 

. in  117  tens  4  tens  (40)  times  with  a  remainder 

175  of  17  tens;  17  tens  equal  170  units;  170  units 

175  plus  5  units  equal  175  units.     Twenty-five  is 

contained  in  175  units  7  times. 
Hence  25  is  contained  in  8675,  347  times. 

132.  Problems. 

1.  93492^49  5.  5904^328 

2.  92169-^77  6.  7693-^  157 

3.  72855-^45  7.  8190-^-546 
4  34694-^38  8.  12960  -^864 

(a)  Find  the  sum  of  the  eight  quotients. 


117 
100 


194  COMPLETE   ARITHMETIC. 

Division— Decimals. 

133.  Find  the  quotient  of  785.65  divided  by  .5. 
Operation.  Explanation. 

.5)785.6^5  First  place  a  separatrix  (v)  after  that  figure  ii\ 

1  ^rr-i  o     the  dividend  that  is  of  the  same  denomination  aa 

the  right-hand  figure  of  the  divisor — in  this  case 

after  the  figure  6.     Then  divide,  writing  the  decimal  point  in  the 

quotient  when,  in  the  process  of  division,  the  separatrix  is  reached — 

in  this  case  after  the  figure  1. 

It  was  required  to  find  how  many  times  5  tenths  are  contained  in 
7856  tenths.  5  tenths  are  contained  in  7856  tenths  1571  times. 
There  are  yet  15  hundredths  to  be  divided.  5  tenths  are  contained 
in  15  tenths  3  times;  in  15  hundredths  3  tenths  of  a  time. 

Note. — By  holding  the  thought  for  a  moment  upon  that  part  of  the  dividend 
which  corresponds  in  denomination  to  the  divisor,  the  place  of  the  decimal  point 
becomes  apparent. 

5  apples  are  contained  in  7856  apples  1571  times. 
5  tenths  are  contained  in  7856  tenths  1571  times. 

134.  Solve  and  explain  the  following  problems  with  special 
reference  to  the  placing  of  the  decimal  point : 

1.  Divide  340  by  .8  .8)340.0^ 

2.  Divide  468.5  by  .25  .25)468.50' 

3.  Divide  38.250  by  12.5  12.5)38.2^50 

4.  Divide  87  by  2.5  2.5)87.0' 

5.  Divide  546  by  .75  .75)546.00' 

6.  Divide  .576  by  2.4  2.4).5'76 

7.  86  ^  .375  =  8.  94.5  ^  .8  = 

9.  75  ^  .15=  10.  125  -^  .5  = 

11.  12.5  ^  .05  =  12.  1.25  ^  .5  = 


(a)  Find  the  sum  of  the  twelve  quotients. 


PART    II. 


195 


Division— United  States  Money. 
135.  Divide  $754.65  by  $.27. 


Operation. 

$.27)$754.65X2795 
54 

214 
189 


256 
243 


135 

135 


Explanation- 

This  means,  find  how  many  times  27 
cents  are  contained  in  75465  cents.  27 
cents  are  contained  in  75465  cents,  2795 
times. 

Problem. 

At  27^  a  bushel,  how  many  bushels  of 
oats  can  be  bought  for  f 754. 65?  As 
many  bushels  can  be  bought  as  $.27  is 
contained  times  in  $754.65.  It  is  con- 
tained 2795  times. 


136.  Divide  $754.65  by  27. 
Operation. 

27)$754.^65($27.^95 
54 

214 
189 


256 
243 


135 
135 


Explanation. 

Tljis  means,  find  one  27th  of  $754.65. 
One  27th  of  $754.65  is  $27.95. 

Note. — One  might  find  1  27th  of 
$754.65  by  finding  how  many  times  $27 
is  contained  in  $754.65. 

Problem. 

If  27  acres  of  land  are  worth  $754,65, 
how  much  is  one  acre  worth? 


137.  Divide  $754.65  by  .27. 
Operation. 

.27)$754.65'($2795 
54 


214 
189 

256 
243 


135 
135 


Explanation. 

This  means,  find  100  27ths  of  $754.65. 
One  27th  of  $754.65  is  $27.95.  100  27ths 
of  $754.65  is  $2795. 

Note. — In  practice,  we  find  one  27th 
of  100  times  $754.65. 

Problem. 

If  .27  of  an  acre  of  land  is  worth 
$754.65,  how  much  is  1  acre  worth. at 
the  same  rate? 


196  COMPLETE    ARITHMETIC. 

Division— Denominate  Numbers. 

138.  Divide  46  rd.  12  ft.  8  in.  by  4. 

Operation.  Explanation. 

4)46  rd.    12  ft.    8  in.         This  means,  find  1  fourth  of  46  rd.  12  ft. 

11  rd.  lift.  5  in.       ^^'      „      ^^    i..«    ,  •    n    ^      -.i 

One  fourth  of  46  rd.  is  11  rd.  with  a  re- 
mainder of  2  rd. ;  2  rd.  equal  33  ft.;  33  ft.  plus  12  ft.  equal  45  ft. 

One  fourth  of  45  ft.  equals  11  ft.  with  a  remainder  of  1  ft. ;  1  ft. 
eouals  12  in. ;  12  in.  plus  8  in.  equals  20  in. 
One  fourth  of  20  in.  equals  5  in. 
One  fourth  of  46  rd.  12  ft.  8  in.  equals  11  rd.  11  ft.  5  in. 

PROBLEM. 

The  perimeter  of  a  square  garden  is  46  rd.  12  ft.  8  in.  How  far 
across  one  side  of  it? 

139.  Miscellaneous. 

Tell  the  meaning  of  each  of  the  following,  solve,  explain,  and 
state  in  the  form  of  a  problem  the  conditions  that  would  give  rise  to 
each  number  process : 

1.  Multiply  64  rd.  14  ft.  6  in.  by  8. 

2.  Divide  37  rd.  15  ft.  4  in.  by  5. 

3.  Divide  $675.36  by  $48. 

4.  Divide  $675.36  by  48. 

5.  Divide  $675.36  by  .48. 

6.  Divide  $675.36  by  $4.8. 

7.  Divide  $675.36  by  4.8. 

8.  Divide  $675.36  by  $.48. 

9.  Multiply  $356.54  by  .36. 

10.  Multiply  $356.54  by  3.6. 

11.  Multiply  $356.54  by  36. 

12.  Can  you  multiply  by  a  number  of  dollars  ? 

13.  Can  you  divide  by  a  number  of  dollars? 


PART    II.  197 

Algebraic  Division. 

140.  Examples. 

No.  1.  No.  2. 

4)12  +  5x4-8  a)12a  +  2ah  +  4a 

3    +    5-2  12  +  2&  +  4 

No.  3.  No.  4. 

2)2^  +  3  X  2'-^  -  2  h)ah'  +  c^'^  +  Sh 

2^  +  3  X  2  -  1    .  a&^  +  c&  +  3 

1.  Prove  Nos.  1  and  3,  by  (1)  reducing  each  dividend  to 
its  simplest  form,  (2)  dividing  it  so  reduced,  by  the  divisor, 
and  (3)  comparing  the  result  with  the  quotient  reduced  to 
its  simplest  form. 

2.  Verify  No.  2  by  letting  a  =  3,  and  b  =  5. 

3.  Verify  No.  4  by  letting  a  -  3,b  =  5,  and  c  =  7. 

141.  (6xa,XctxaxaxcL)^(2xaxci)  =  6a^-f-  2^^=  3al 

Observe  that  to  divide  one  algebraic  term  by  another  we 
must  find  the  quotient  of  the  coefficients  and  the  difference  of 
the  exponents. 

142.  Problems. 

1.  6a'b  -^2a=  3.  Sa'b'  -^  2a  = 

2.  4:a*b'  ^2a=  4.  lOd'b'  -^  2a  =^ 

5. 

2a)6a'b  +  4aV  -  8a'b'  +  lOa'b' 

6.  Verify  problem  5  by  letting  a  =  S  and  &  =  5. 


198  COMPLETE    ARITHMETIC. 

Algebraic  Division. 
143.  Problems. 

1.  Divide  4:a^x  +  8a V  +  6ax^  by  2ax. 

2.  Multiply  the  quotient  of  problem  1  by  2ax. 

3.  Verify  problems  1  and  2  by  letting  a  =  2  and  x  =  S. 

4.  Divide  Sah'  +  6aV  +  9a%  by  3ah. 

5.  Multiply  the  quotient  of  problem  4  by  Sah. 

6.  Verify  problems  4  and  5  by  letting  a  =  3  and  h  =  5. 

7.  Divide  2x^y  -\-  x?y^  —  xy^  by  xy. 

8.  Multiply  the  quotient  of  problem  7  by  xy. 

9.  Verify  problems  7  and  8  by  letting  x  =  2  and  2/  =  3. 

10.  Divide  5ay  -  2(2'?/'  +  dY  by  ft'?/. 

11.  Multiply  the  quotient  of  problem  10  by  ct'y. 

12.  Verify  problems  10  and  11  by  letting  a  -1  and  y  =  2. 

13.  Divide  Wx  +  h'x'  -  36V  by  &;r. 

14.  Multiply  the  quotient  of  problem  13  by  hx. 

15.  Verify  problems  13  and  14  by  letting  &  =  3  and  x  -  4:. 

Observe  that  when  the  divisor  is  a  positive  number,  each 
term  of  the  quotient  has  the  same  sign  as  the  term  in  the 
dividend  from  which  it  is  derived. 

2)8  -  6       One  half  of  +  8  is  +  4 ;  one  half  of  -  6  is  -  3. 


16.  2x)4x'  -  Qx'  +  8^'  -  2x''  +  6x. 


PART    II. 
Geometry. 

144.  Triangles. 


199 


llight 


Isosceles 


Isosceles  Equilateral 


1.  A  triangle  has 


sides  and 


angles. 


2.  A  right  triangle  has  one  right  angle ;  that  is,  one  angle 
of  degrees. 

3.  An  isosceles  triangle  has  two  angles  that  are  equal  and 
two  sides  that  are  equal. 

4.  An   equilateral   triangle   has    equal    sides    and   equal 
angles. 


Fig.  5 


Fig.  6 


5.  Cut  from  paper  a  triangle  similar  to  the  one  shown  in 
Fig.  5.  Then  cut  it  into  parts  as  shown  by  the  dotted  lines. 
Ee-arrange  the  3  angles  of  the  triangle  as  shown  in  Fig.  6. 
Compare  the  sum  of  the  3  angles  with  two  right  angles  as 
shown  in  Fig.  6.  Convince  yourself  that  the  three  angles  of 
this  triangle  are  together  equal  to  two  right  angles, 

6.  Cut  other  triangles  and  make  similar  compaiisons,  until 
you  are  convinced  that  the  sum  of  the  angles  of  any  triangle 
is  equal  to  two  right  angles. 


200 


COMPLETE    ARITHMETIC. 
145.  Miscellaneous  Review. 


1.  Ji  in  figure  1,  the  angle  <x  is  a  right  angle,  and  the 
angle  h  is  equal  to  the  angle  c,  the  angle  h  is  an  angle  of 
how  many  degrees  ? 

2.  If  in  figure  2,  the  angle  d  is  an  angle  of  95°  and  the 
angle  e  is  an  angle  of  40°,  the  angle  /  is  an  angle  of  how 
many  degrees  ? 

3.  If  in  figure  3,  the  angle  x  is  an  angle  of  75°,  the  angle 
w  is  an  angle  of  how  many  degrees  ? 

4.  If  in  an  oblong  there  are  ah  square  feet,  and  the  oblong 
is  a  feet  long,  it  is feet  wide,     ab  -^  a  = 

5.  If  in  a  rectangular  solid  there  are  ahc  cubic  feet,  and 

the  solid  is  a  feet  long  and  h  feet  wide,  it  is  feet 

thick,     ahc  ^  ah  = 

6.  Verify  problems  4  and  5  by  letting  a  =  3,  &  =  4,  and 
c  =  2. 

7.  There  is  a  field  that  contains  1736  square  rods;  it  is 
28  rods  long.     How  wide  is  the  field  ? 

8.  There  is  a  solid  that  contains  4320  cubic  inches;  it  is 
24  inches  long  and  15  inches  wide.    How  thick  is  the  solid  ? 

9.  How  many  square  inches  of  surface  in  the  solid  de- 
scribed in  problem  8  ? 


PEOPERTIES  OF  NUMBERS. 

To  THE  Teacher. — Under  this  head,  number  in  the  abstract  is 
discussed  with  little  or  no  distinction  between  numbers  of  things 
and  pure  number.  It  is  dissociation  and  generalization,  Mdthout 
wliich  there  could  be  little  progress  in  the  "  science  of  number  "  or 
in  the  "art  of  computation." 

146.  Every  number  is  fractional,  integral,  or  mixed. 

1.  A  fractional  number  is  a  number  of  the  equal  parts  of 
some  quantity  considered  as  a  unit;  as,  |-,  .9,  5  sixths. 

2.  An  integral  number  is  a  number  that  is  not,  either 
wholly  or  in  part,  a  fractional  number;  as,  15,  46,  ninety- 
five. 

3.  A  mixed  number  is  a  number  one  part  of  which  is 
integral  and  the  other  part  fractional;  as,  5|-,  27.6,  274|. 

147.  An  exact  divisor  of  a  number  is  a  number  that  is 
contained  in  the  number  an  integral  number  of  times. 

5  is  an  exact  divisor  of  15. 
5  is  not  an  exact  divisor  of  1.5. 
16|  is  an  exact  divisor  of  100. 

148.  Every  integral  number  is  odd  or  even. 

1.  An  odd  number  is  a  number  of  which  two  is  not  an 
exact  divisor;  as,  7,  23,  141. 

2.  An  even  number  is  a  number  of  which  two  is  an  exact 
divisor;  as,  8,  24,  142. 


201 


202  COMPLETE   ARITHMETIC. 

Properties  of  Numbers. 

149.  Every  integral  number  is  prime  or  composite. 

1.  A  prime  number  is  an  integral  number  that  has  no 
exact  integral  divisors  except  itself  and  1 ;  as,  23,  29,  31,  etc. 

2.  Is  two  a  prime  number  ?  three  ?  nine  ? 

3.  Name  the  prime  numbers  from  1  to  97  inclusive.  Find 
their  sum. 

4.  A  composite  number  is  an  integral  number  that  has 
one  or  more  integral  divisors  besides  itself  and  1 ;  as,  6,  8, 
9,  10,  12,  14,  15,  etc. 

5.  Name  the  composite  numbers  from  4  to  100  inclusive. 
Find  their  sum. 

6.  Is  eight  a  composite  number  ?  eleven  ?  fifteen  ? 

(a)  Find  the  sum  of  the  results  of  problems  No.  3  and 
No.  5. 

150.  To  find  whether  an  integral  number  is  prime  or 
composite. 

1.  Is  the  number  371  prime  or  composite? 

Operation.  Explanation. 

2)371         5)371  Beginning  with  2  (the  smallest  prime 

Qp.  .  7^  I        number  except  the  number  1),  it  is  found 

by  trial  not  to  be  an  exact  divisor  of  371. 
3)371        7)371  3  is  not  an  exact  divisor  of  371. 

123-1-  53  ^  ^^  ^^^^  ^^^  exact  divisor  of  371. 

7  is  an  exact  divisor  of  371.  Therefore 
371  is  a  composite  number,  being  composed  of  53  sevens,  or  of  7 
fifty-threes. 

Observe  that  we  use  as  trial  divisors  only  prime  numbers. 
If  2  is  not  an  exact  divisor  of  a  number,  neither  4  nor  6  can 
be.     Do  you  see  why  ? 


PART   II.  203 

Properties  of  Numbers. 

2.  Is  the  number  397  prime  or  composite  ? 

Operation.  Explanation. 

2)397  3)397  By  trial  it  is  found  that  neither  2,  3, 

-,  QQ  -,  Qo  ,         5,  7, 11, 13, 17,  nor  19  is  an  exact  divisor 

0)oy7  /joy^  ;f^Q  composite  number  between  2  and 

79_|_  56-}-        19  can  be  an  exact  divisor  of  397;  for 

11^SQ7         1^'>SQ7  since  one  2  is  not  an  exact  divisor  of  the 

number,  several  2's,  as  4,  6,  8,  12,  etc., 

OD-f-  6\)-f-        cannot  be ;  since  one  3  is  not  an  exact 

17)397         19)397  divisor  of  the  number,  several  3's,  as  6,  9, 

2S-I-  20-1         ^'^'>  ^^^*'  ^^^^^^ot  be ;  since  one  5  is  not  an 

exact  divisor  of  the  number,  several  5's, 

as  10  and  15,  cannot  be;  since  one  7  is  not  an  exact  divisor  of  the 

number,  two  7's  (14)  cannot  be. 

No  number  greater  than  19  can  be  an  exact  divisor  of  the  number ; 
for  if  a  number  greater  than  19  were  an  exact  divisor  of  the  number, 
the  quotient  (which  also  must  be  an  exact  divisor)  would  be  less  than 
20.  But  it  has  aheady  been  proved  that  no  integral  number  less 
than  20  is  an  exact  divisor  of  397.  Therefore  397  is  a  prime  number. 

Observe  that  in  testing  a  number  to  determine  whether  it 
is  prime  or  composite,  we  take  as  trial  divisors  prime  num- 
bers only,  beginning  with  the  number  two. 

Observe  that  as  the  divisors  become  greater,  the  quotients 
become  less,  and  that  we  need  make  no  trial  by  which  a  quo- 
tient will  be  produced  that  is  less  than  the  divisor. 

3.  Determine  by  a  process  similar  to  the  foregoing 
whether  each  of  the  following  is  prime  or  composite:  127, 
249,  257,  371. 

151.  Any  divisor  of  a  number  may  be  regarded  as  a  fac- 
tor of  the  number.  An  exact  integral  divisor  of  a  number 
is  an  integral  factor  of  the  number. 


204  COMPLETE    ARITHMETIC. 

Properties  of  Numbers. 
152.  Prime  Factors. 

1.  An  integral  factor  that  is  a  prime  number  is  a  prime 
factor. 

5  is  a  prime  factor  of  30. 

7  is  a  prime  factor  of  and  . 

3  is  a  prime  factor  of  and  . 

2  and  3  are  prime  factors  of  and  and  . 

3  and  5  are  prime  factors  of  and  and 

2.  Eesolve  105  into  its  prime  factors. 

Operation.  Explanation. 

5)105  Since  the  prime  number  5  is  an  exact  divisor  of  105, 

Qwi         it  is  a  prime  factor  of  105.     Since  tiie  prime  number  3 

' — ^        is  an  exact  divisor  of  the  quotient  (21),  it  is  a  prime 

factor  of  21  and  105. 

Since  3  is  contained  in  21  exactly  7  times,  and  since  7  is  a  prime 

number,  7  is  a  prime  factor  of  21  and  of  105.     Therefore  the  prime 

factors  of  105  are  5,  3,  and  7. 

Observe  that  if  7  and  3  are  prime  factors  of  21  they  must 
be  prime  factors  of  105,  for  105  is  made  up  of  5  21's.  7  is 
contained  5  times  as  many  times  in  105  as  it  is  in  21. 

Observe  that  every  composite  number  is  equal  to  the 
product  of  its  prime  factors. 

105-5x3x7.     18=:3x3x  2. 

Observe  that  2  times  3  times  a  number  equals  6  times  the 
number;  3  times  5  times  a  number  equals  15  times  the 
number,  etc. 

Observe  that  instead  of  multiplying  a  number  by  21,  it 
may  be  multiplied  by  3  and  the  product  thus  obtained  by  7, 
and  the  same  result  be  obtained  as  would  be  obtained  by 
multiplying  the  number  by  21.     Why? 


PART    II.  205 

Properties  of  Numbers. 

153.  Multiples,  Common  Multiples,  and  Least  Common 
Multiples. 

1.  A  multiple  of  a  number  is  an  integral  number  of  times 
tlie  number. 

30  and  35  and  40  are  multiples  of  5. 

16  and  20  and  32  are  multiples  of  4.  • 

2.  A  common  multiple  of  two  or  more  numbers  is  an  in- 
tegral number  of  times  each  of  the  numbers. 

30  is  a  common  multiple  of  5  and  3. 
40  is  a  common  multiple  of  8  and  10. 

is  a  common  multiple  of  9  and  6. 

is  a  common  multiple  of  8  and  12. 

3.  A  common  multiple  of  two  or  more  integral  numbers 
contains  all  the  prime  factors  found  in  every  one  of  the  nnm- 
bers,  and  may  contain  other  prime  factors. 

48  =  2x2x2x2x3.  150  =  2x3x5x5.  A  common  mul- 
tiple of  48  and  150  must  contain  four  2's,  one  3,  and  two  5's.  It 
may  contain  other  factors. 

2x2x2x2x3x5x5  =  1200. 
2x2x2x2x3x5x5x2  =  2400. 
1200  and  2400  are  common  multiples  of  48  and  150. 

4.  The  least  common  multiple  (1.  c.  m.)  of  two  or  more 
numbers  is  the  least  number  that  is  an  integral  number  of 
times  each  of  the  numbers. 

40  and  80  and  120  are  common  multiples  of  8  and  10 ;  but  40  is 
the  least  common  multiple  of  8  and  10. 

5.  The  least  common  multiple  of  two  or  more  numbers 
contains  all  the  prime  factors  found  in  every  one  of  the  num- 
bers, and  no  other  prime  factors. 


206  COMPLETE    ARITHMETIC. 

Properties  of  Numbers. 

36  =  2  X  2  X  3  X  3.  120  =  2  X  2  X  2  X  3  X  5.  The  1.  c.  m.  of 
36  and  120  must  contain  three  2's,  two  3's,  and  one  5.  2  X  2  X  2  X 
3x3x5  =  360,  1.  c.  m.  of  36  and  120. 

6.  To  find  the  1.  c.  m.  of  two  or  more  numbers:  Resolve 
each  number  into  Us  prime  factors.  Take  as  factors  of  the 
I.  c.  m.  the  greatest  number  of  2's,  3's,  5's,  7's,  etc.,  found  in 
any  one  of  the  numbers. 

Example. 
Find  the  1.  c.  m.  of  24,  35,  36,  and  50. 

Operation. 
24  =  2x2x2x3. 

35  =  5  X  7.  . 

36  =  2x2x3x3. 
50  =  2  X  5  X  5. 

2x2x2x3x3x5x5x7  =  12600, 1.  c.  m. 

Explanation. 

24  has  the  greatest  number  of  2's  as  factors. 

36  has  the  greatest  number  of  3's  as  factors. 

50  has  the  greatest  number  of  5's  as  factors. 

35  is  the  only  number  in  which  the  factor  7  occurs. 
There  must  be  as  many  2's  among  the  factors  of  the  1.  c.  m.  as 
there  are  2's  among  the  factors  of  24 ;  as  many  3's  as  there  are  3's 
among  the  factors  of  36 ;  as  many  5's  as  there  are  5's  among  the 
factors  of  50 ;  as  many  7's  as  there  are  7's  among  the  factors  of  35 ; 
that  is,  three  2's,  two  3's,  two  5's,  and  one  7. 

Find  the  1.  c.  m.: 

7.  Of  48  and  60.  11.  Of  20,  30,  and  40. 

8.  Of  60  and  75.  12.  Of  40,  50,  and  60. 

9.  Of  50  and  60.  13.  Of  24,  48,  and  36. 
10.  Of  30  and  40.  14.  Of  25,  35,  and  40. 


PART  n.  ^07 

Algebra — Parentheses. 

154.  When  an  expression  consisting  of  two  or  more  terms 
is  to  be  treated  as  a  whole,  it  may  be  enclosed  in  a  paren- 
thesis. 

(  12  +  (5  +  3)  =  ?  (  7a  +  (3a  +  2a)  =  ? 

I  12  +  5  +  3  =  ?  (  7a  +  3a  +  2a  .--  ? 

Observe  that  removing  the  parenthesis  makes  no  change  in  the 
results. 

M  2  _  (5  +  3)  =  ?  (  7a  -  (3a  +  2a)  =  ? 

(  12  -  5  -  3  =  ?  (  7a  -  3a  -  2a  =  ? 

Observe  the  change  in  signs  made  necessary  by  the  removal  of  the 
parenthesis. 

(  12  -  (5  -  3)  =  ?  I  7a  -  (3a  -  2a)  =  ? 

(12-5  +  3  =  ?  (7a-3a  +  2a  =  ? 

Observe  the  change  in  signs  made  necessary  by  the  removal  of  the 
parenthesis. 

A  careful  study  and  comparison  of  the  foregoing  prob- 
lems will  make  the  reasons  for  the  following  apparent : 

I.  If  an  expression  within  a  parenthesis  is  preceded  by  the  plus 
sign,  the  parenthesis  may  be  removed  without  making  any  changes 
in  the  signs  of  the  terms. 

II.  If  an  expression  within  a  parenthesis  is  preceded  by  a  minus 
sign,  the  parenthesis  may  be  removed;  but  the  sign  of  each  termin  the 
parenthesis  must  be  changed ;  the  sign  +  to  - ,  and  the  sign  -  to  +. 

155.  Eemove  the  parenthesis,  change  the  signs  if  neces- 
sary, and  combine  the  terms  : 

1.  15  _  (6  -f.  4)  =  5.  15b  -  {12b  -  46)  = 

2.  18  +  (4  -  3)  =  6.  18c  +  (9c  -  3c)  = 

3.  27-(8  +  3)=  7.  24c^-(5^+3^)  = 

4.  45 +  (12 -3)=  8.  36a;-(5aj  +  4ic)  = 


208  COMPLETE    ARITHMETIC. 

Algebra — Parentheses. 

156.  Multiplying  an  Expression  Enclosed  in  a 
Parenthesis. 

1.  6(7  +  4)  =-  ?*  6(7a  +  4&)  =  ? 

Q(a  -I-  Z?)  =  ?     Ans.  6a  +  6b. 
a(b  +  c)  =  ?     Ans.  ah  +  ac. 
Observe  that  in  multiplying  the  sum  of  two  numbers  by  a  third 
number,  the  sum  may  be  found  and  multiplied;    or  each  number 
may  be  multiplied  and  the  sum  of  the  products  found. 

In  the  last  three  examples  given  above,  substitute  5  for  a, 
3  for  h,  and  2  for  c  ;  then  perform  again  the  operations  indi- 
cated, and  compare  the  results  with  those  obtained  when  the 
letters  were  employed. 

2.  6(7  -  4)  =  ?  6(7a  -  45)  =  ? 

6(a  —  h)  =  1     Ans.  6a  —  6b. 

a(b  —  c)  =  ?     Ans.  ah  —  ac. 
Observe  that  in  multiplying  the  difference  of  two  numbers  by  a 
third  number,  the  difference  may  be  found  and  multiplied ;  or  each 
number  may  be  multiplied  and  the  dilference  of  the  products  found. 

In  the  last  three  problems  given  above,  substitute  5  for  a, 
3  for  b,  and  2  for  c;  then  perform  again  the  operations  indi- 
catedi  and  compare  the  results  with  those  obtained  when  the 
letters  were  employed. 

157.  Problems. 
li  a  =  6,b  =  S,  and  c  =  2,  find  the  value  of  the  following : 

1.  S(a  +  b)-2(b-\-c). 

2.  4(a  +  2b)  -  3(6  -  c). 

3.  2{2a  -b)  +  2(2h  -  c). 

*  This  means,  that  the  sum  of  7  and  4  is  to  be  multiplied  by  six  ;  or  that  the  sum 
of  six  7's  and  six  4's  is  to  be  found. 


PART    It. 

Geometry. 

158.  Triangles — Continued. 


209 


Right  Triangle. 

1.  The  sum  of  the  angles  of  any  triangle  is  equal  to  

right  angles  or  degrees. 

2.  In  a  right  triangle  there  is  one  right  angle.  The  other 
two  angles  are  together  equal  to . 

3.  In  a  certain  right  triangle  one  of  the  angles  is  an  angle 
of  40°.  How  many  degrees  in  each  of  the  other  two  angles  ? 
Draw  such  a  triangle. 

4.  Convince  yourself  by  drawings  and  measurements  that 
every  equilateral  triangle  is  equiangular. 


Equilateral 
Triangles. 


Equiangular 
Triangles. 


5.  Note  that  in  every  equiangular  triangle  each  angle  is 
one  third  of  2  right  angles.  So  each  angle  is  an  angle  of 
degrees. 

6.  If  any  one  of  the  angles  of  a  triangle  is  greater  or  less 
than  60,  can  the  triangle  be  equiangular?  a 
Can  it  be  equilateral  ? 

7.  If  angle  a  of  an  isosceles  triangle  meas- 
ures 50°,  how  many  degrees  in  angle  &? 
In  angle  c  ? 


210  COMPLETE    ARITHMETIC. 

159.  Miscellaneous  Review. 

1.  I  am  thinking  of  a  right  triangle  one  of  whose  angles 
measures  32°.  Give  the  measurements  of  the  other  two 
angles.     Draw  such  a  triangle. 

2.  I  am  thinking  of  an  isosceles  triangle ;  the  sum  of  its 
two  equal  angles  is  100°.  Give  the  measurement  of  its 
third  angle.     Draw  such  a  triangle. 

3.  Let  a  equal  the  number  of  degrees  in  one  angle  of  a 
triangle  and  h  equal  the  number  of  degrees  in  another  angle 
of  the  same  triangle;  then  the  number  of  degrees  in  the 
third  angle  is  180°  —  (a  +  ^)-  If  (^  equals  30,  and  h  equals 
45,  how  many  degrees  in  the  third  angle  ? 

4.  Name  three  common  multiples  of  16  and  12. 

5.  Name  the  hast  common  multiple  of  16  and  12. 

6.  Find  the  sum  of  all  the  prime  numbers  from  101  to 
127  inclusive. 

7.  Find  the  prime  factors  of  836. 

8.  With  the  prime  factors  of  836  in  mind  or  represented 
on  the  blackboard,  tell  the  following : 

{(£)  How  many  times  is  19  contained  in  836  ? 

(6)  How  many  times  is  11  x  19  contained  in  836  ? 

(c)  How  many  times  is  19  x  11  x  2  contained  in  836? 

160.  Problems. 
Find  the  L  c.  m. 

1.  Of  18  and  20.  6.  Of  36,  72,  and  24. 

2.  Of  13  and  11.  7.  Of  45,  81,  and  27. 

3.  Of  24  and  32.  8.  Of  33,  55,  and  88. 

4.  Of  16  and  38.  9.  Of  45,  65,  and  85. 

5.  Of  46  and  86.  10.  Of  3,  5,  7,  and  11. 
(a)  Find  the  sum  of  the  ten  results. 


DIVISIBILITY  OF  NUMBEES. 

161.  Numbers  Exactly  Divisible  by  2;    by  2|;    by  3J; 
BY  5;   by  10. 

1.  An  integral  number  is  exactly  divisible  by  2  if  the 
right-hand  figure  is  0,  or  if  the  number  expressed  by  its 
right-hand  figure  is  exactly  divisible  by  2. 

Explanatory  Note. — Every  integral  number  that  may  be 
expressed  by  two  or  more  figures  may  be  regarded  as  made  up  of  a 
certain  number  of  tens  and  a  certain  number  (0  to  9)  of  primary 
units;  thus,  485  is  made  up  of  48  tens  and  5  units ;  4260  is  made  up 
of  426  tens  and  0  units ;  27562  is  made  up  of  2756  tens  and  2  units. 
But  ten  is  exactly  divisible  by  2  ;  so  any  number  of  tens,  or  any 
number  of  tens  plus  any  number  of  twos,  is  exactly  divisible  by  2. 

2.  Tell  which  of  the  following  are  exactly  divisible  by  2, 
and  why:  387,  5846,  2750,  2834. 

3.  Any  number,  integral  or  mixed,  is  exactly  divisible  by 
2^-  if  the  part  of  the  number  expressed  by  figures  to  the 
right  of  the  tens'  figure,  is  exactly  divisible  by  2^. 

4.  Show  why  the  statement  made  in  No.  3  is  correct, 
employing  the  thought  process  given  in  the  "  Explanatory 
Kote"  above. 

5.  Tell  which  of  the  following  are  exactly  divisible  by'2^, 
and  why:  485,  470,  365,  472J,  3847^. 

6.  Any  number,  integral  or  mixed,  is  exactly  divisible 
by  31  if  the  part  of  the  number  expressed  by  figures  to  the 
right  of  the  tens'  figure  is  exactly  divisible  by  3i-. 

7.  Tell  which  of  the  following  are  exactly  divisible  by 
^,  and  why:  780,  2831,  576f,  742,  80. 

211 


212  COMPLETE    ARITHMETIC. 

Divisibility  of  Numbers. 

8.  Any  integral  number  is  exactly  divisible  by  5  if  it: J 
right-hand  figure  is  0  or  5.     Show  why. 

9.  Any  integral  number  is  exactly  divisible  by  10  if  ii% 
right-hand  figure  is  . 

162.  Problems. 

1.  How  many  times  is  2i  contained  in  582^?* 

2.  How  many  times  is  2|^  contained  in  375  ? 

3.  How  many  times  is  2^  contained  in  467|-? 

4.  How  many  times  is  2i  contained  in  4680  ? 

5.  How  many  times  is  3  J  contained  in  786|?t 

6.  How  many  times  is  3^  contained  543^? 

7.  How  many  times  is  3i  contained  in  8640? 

8.  How  many  times  is  5  contained  in  3885? 

9.  How  many  times  is  5  contained  in  1260? 

163.  Numbers  Exactly  Divisible  by  25;  by  33^;  by  1  2|-; 
BY  16|;  BY  20;  by  50. 
1.  Any  integral  number  is  exactly  divisible  by  25  if  its 
two  right-hand  figures  are  zeros,  or  if  the  part  of  the  number 
expressed  by  its  two  right-hand  figures  is  exactly  divisible 
by  25. 

Explanatory  Note. — Every  integral  number  expressed  by  three 
or  more  figures  may  be  regarded  as  made  up  of  a  certain  number  of 
hundreds  and  a  certain  number  (0  to  99)  of  primary  units;  thus  4624 
is  madje  up  of  46  hundreds  and  24  units ;  38425  is  made  up  of  384 
hundreds  and  25  units;  8400  is  made  up  of  84  hundreds  and  0 
units.  But  a  hundred  is  exactly  divisible  by  25 ;  so  any  number  of 
hundreds,  or  any  tiumber  of  hundreds  plus  any  number  of  25's  is 
exactly  divisible  by  25. 

*  2J  is  contained  in  582}  (4  X  58)  + 1  times.    Why  ? 
1 3i  is  contained  in  786|  (3  X  78)  +  2  times.    Why  ? 


PART   II.  213 

Divisibility  of  Numbers. 

2.  Tell  which  of  the  following  are  exactly  divisible  by 
25,  and  why :  37625,  34836,  27950,  38575. 

3.  Every  number,  integral  or  mixed,  is  exactly  divisible 
by  33i,  if  that  part  of  the  number  expressed  by  the  figures 
to  the  right  of  the  hundreds'  figure  is  exactly  divisible  by  33^. 

4.  Show  why  the  statement  made  in  No.  3  is  correct, 
employing  the  thought  process  given  in  the  "  Explanatory 
Note  "  under  No.  1  on  the  preceding  page. 

5.  Tell  which  of  the  following  are  exactly  divisible  by 
33i,  and  why:  36466|,  2375,  46833^  38900,  46820. 

6.  Any  number,  integral  or  mixed,  is  exactly  divisible  by 
12|^,  if  the  part  of  the  number  expressed  by  the  figures  to 
the  right  of  the  hundreds'  figure,  is  exactly  divisible  by  12|^. 
Show  why. 

7.  Tell  which  of  the  following  are  exactly  divisible  by 
12^,  and  why:  375,  837^,  6450,  4329,  7467^. 

8.  Any  number,  integral  or  mixed,  is  exactly  divisible  by 
16|,  if 

9.  Tell  which  of  the  following  are  exactly  divisible  by 
16f :  46331,  5460,  2350,  37400,  275831  2541 6f. 

10.  Any  integral  number  is  exactly  divisible  by  20  if  the 
number  expressed  by  its  two  right-hand  figures  is  exactly 
divisible  by  20.     Show  why. 

11.  Tell  which  of  the  following  are  exactly  divisible  by 
20,  and  why:  3740,  2650,  3860,  29480,  3470. 

12.  Tell  which  of  the  following  are  exactly  divisible  by 
50,  and  why :  2460,  3450,  6800,  27380,  25450. 


214  COMPLETE    ARITHMETIC. 

Divisibility  of  Numbers. 
164.  Problems. 

1.  How  many  times  is  25  contained  in  2450  ?* 

2.  How  many  times  is  25  contained  in  3775  ? 

3.  How  many  times  is  33^  contained  in  4666f  ?  f 

4.  How  many  times  is  33|-  contained  in  343 3^  ? 

5.  How  many  times  is  12^-  contained  in  4737 1^  ? 

6.  How  many  times  is  12-|-  contained  in  3662|^  ? 

7.  How  many  times  is  16|  contained  in  2533^? 

8.  How  many  times  is  16|  contained  in  4550  ? 

165.  Numbers  Exactly  Divisible  by  9. 

1.  Any  number  is  exactly  divisible  by  9  if  the  sum  of  its 
digits  is  exactly  divisible  by  9. 

Explanatory  Note. — Any  number  more  than  nine  is  a  certain 
number  of  nines  and  as  many  over  as  the  number  indicated  by  the 
sum  of  its  digits.  Thus,  20  is  two  nines  and  2  over;  41  is  four 
nines  and  4  -|-  1  over;  42  is  four  nines  and  4  -f  2  over;  200  is 
twenty-two  nines  and  2  over ;  300  is  thirty-three  nines  and  3  over ; 
320  is  a  certain  number  of  nines  and  3  -|-  2  over;  321  is  a  certain 
number  of  nines  and  3  -f  2  -f- 1  o/er. 

326  is  a  certain  number  of  nines  and  3  -|-  2  +  6  over;  but  3  -|- 
2  -f-  6  =  11,  or  another  nine  and  2  over. 

2.  Eead  the  "  Explanatory  Note  "  carefully,  and  tell  which 
of  the  following  are  exactly  divisible  by  9 :  3256,  4266, 
2314,  2574. 

166.  Problems. 

1.  4625  is  a  certain  number  of  9's  and over. 

2.  3526  is  a  certain  number  of  9's  and  over. 

3.  2154  is  a  certain  number  of  9's  and over. 

*  25  is  contained  in  2450  (4  x  24)  +  2  times.    Why? 
+  33J  is  contained  in  46661  (3  x  46)  -|-  2  times.    Why  ? 


PART    II.  215 

Divisibility  of  Numbers. 
167.  Prime  Eactoks  and  Exact  Divisors. 

1.  Any  integral  number  is  exactly  divisible  by  each  of  its 
prime  factors  and  by  the  product  of  any  two  or  more  of  its 
prime  factors.  Thus,  30, (2x3x5),  is  exactly  divisible  by 
2,  by  3,  by  5,  and  by  (2  x  3),  6,  and  by  (2  x  5),  10,  and  by 
(3  X  5),  15. 

2.  The  exact  integral  divisors  of  36,  (2  x  2  x  3  x  3),  are 
2,  3,  — ,  — ,  — ,  and  . 

168.  Prime  E actors'.  Common  Divisors,  and  Greatest 
Common  Divisors. 

1.  Any  prime  factor  or  any  product  of  two  or  more  prime 
factors  common  to  two  or  more  numbers  is  a  common  divisor 
of  the  numbers.  Thus,  the  numbers  30,  (2  x  3  x  5),  and  40, 
(2x2x2x5),  have  the  factors  2  and  5  in  common.  So 
the  common  divisors  of  30  and  40  are  2,  5,  and  10,  and  the 
greatest  common  divisor  is  10. 

Eule. — To  Jind  the  greatest  common  divisor  of  two  or  more 
numbers,  find  the  product  of  the  prime  factors  common  to  the 
numbers. 

2.  Eind  the  g.  c.  d.  of  50,  75,  and  125. 


Operation  No.  1. 

Opei 

-ation  No.  2. 

50  -  2  X  5  X  5. 

5 

50 

75          125. 

75  =  3  X  5  X  5. 
125  =  5  X  5  X  5. 

5 

10 

15            25. 

2 

3              5 

5  X  5  =  25,  g.  c.  d. 

5  X 

5  =  25,  g.  c.  d 

3. 

Eind  the  g.  c.  d.  of  80, 

IOC 

),  140. 

4. 

Eind  the  g.  c.  d.  of  48, 

60, 

72 

5.  Eind  the  g.  c.  d.  of  64,  96,  256. 


216  COMPLETE    ARITHMETIC. 

Divisibility  of  Numbers. 

6.  Find  the  g.  c.  d.  of  640  and  760. 

Operation.  Explanation. 

640)760(1  The  number  760  is  an  integral  number 

640  of  times  the  g.  c.  d.,  whatever  that  may  he ; 

T^x/^^Q/r  90  is  the  number  640.     We  make  an  in- 

600  complete  division  of  760  by  640  and  have 

as  a  remainder  the  number  120.     Since 

40)120(3      640  and  760  are  each  an  integral  number 
1^0  of  times  the  g.  c.  d.,  their  difference,  120, 

must  be  an  integral  number  of  times  the 
g.  c.  d. ;  for,  taking  an  integral  number  of  times  a  thing  from  an 
integral  number  of  times  a  thing  must  leave  an  integral  number  of 
times  the  thing.  Therefore,  no  number  greater  than  120  can  be  the 
g.  c.  d.  But  if  120  is  an  exact  divisor  of  640,  it  is  also  an  exact 
divisor  of  760,  for  it  will  be  contained  one  more  time  in  760  than  in 
640.  We  make  the  trial,  and  find  that  120  is  not  an  exact  divisor  of 
640 ;  there  is  a  remainder  of  40.  Since  600  (120  X  5)  and  640  are 
each  an  integral  number  of  times  the  g.  c.  d.,  40  nmst  be  an  integral 
number  of  times  the  g.  c.  d.  But  if  40  is  an  exact  divisor  of  120  it 
is  an  exact  divisor  of  600  (120  X  5)  and  640  (40  more  than  600) 
and  760  (120  more  than  640).  We  make  the  trial,  and  find  that  it 
is  an  exact  divisor  of  120,  and  is  therefore  the  g.  c.  d.  of  640  and  760. 
Observe  that  any  number  that  is  an  exact  divisor  of  two  numbers  is 
an  exact  divisor  of  their  difference. 

169.  From  the  foregoing  make  a  rule  for  finding  the  g.  c.  d. 
of  two  numbers  and  apply  it  to  the  following : 
Find  the  g.  c.  d. : 

1.  Of  380  and  240.  6.  Of  540  and  450. 

2.  Of  275  and  155.  7.  Of  320  and  860. 

3.  Of  144  and  96.  8.  Of  475  and  350. 

4.  Of  1728  and  288.  9.  Of  390  and  520. 

5.  Of  650  and  175.  10.  Of  450  and  600. 
(a)  Find  the  sum  of  the  ten  results. 


PART    II.  217 

Algebra— Equations. 

170.  An  equation  is  the  expression  of  the  equality  of  two 
numbers  or  combinations  of  numbers. 

Equations. 

(1)  2+4+6  =  3  +  5  +  4 

(2)  a  +  h  +  c  =  40  -12 

1.  Every  equation  is  made  up  of  two  members.  The  part 
of  the  equation  which  is  on  the  left  of  the  sign  of  equahty 
is  called  the  first  member ;  the  part  on  the  right  of  the  sign 
of  equality,  the  second  member. 

2.  If  the  same  number  be  added  to  each  member  of  an 
equation,  the  equality  will  not  be  destroyed. 

If  X  =  8,  then  ^'  +  4  =  8  +  4. 

li  a  -\-b  =  16,  then  a  +  &  +  c  =  16  +  c. 

3.  If  the  same  number  be  subtracted  from  each  member 
of  an  equation,  the  equality  will  not  be  destroyed. 

If  ic  =  8,  then  x  -  3  =  S  -  3. 

If  a  +  &  =  16,  then  a-\-b  —  c  =  16  —  c. 

4.  If  each  member  of  an  equation  be  multiplied  by  the 
same  number,  the  equality  will  not  be  destroyed. 

If  ic  =  8,  then  4x  =  4:  times  8,  or  32. 

If  a  +  &  =  16,  then  4a  +  4&  =  4  times  16,  or  64. 

5.  If  each  member  of  an  equation  be  divided  by  the  same 
number,  the  equality  will  not  be  destroyed. 

If  ^  =  8,  then  —  =  — ,  or  2. 
4        4 

If  a  +  6  =  16,  then  — H =  — ,  or  4. 

4       4       4 


218 


COMPLETE    ARITHMETIC. 


Algebra— Equations. 

7.  Any  term  in  an  equation  may  be  transposed  from  one 
member  of  the  equation  to  the  other ;  but  its  sign  must  be 
changed  when  the  transposition  is  made. 

If  ic  +  5  =  15,  then  x  =  15  -  5,  or  10. 

liy  -6  =  27,  then  y  =  27  +  6,  or  33. 

li  a -i-h-\-c  =  18,  then  a  -\-  h  =  18  -  c. 

li  x-\-y  —  z  =  2b,  then  x  -\- y  =  25  -\- z. 

171.  To  Find  the  Number  for  Which  x  Stands  in  an 
Equation  in  Which  There  Is  No  Other  Unknown 
Number. 

Example  No.  1. 

Equation,         x  -\-  2x  -\-  ?>x  —  5  =  Vd 
Transposing,  x  -\- 2x -{- ^  x  =  IZ  -\-  5 

Uniting,  6^  =  18 

Dividing,  x  =  ?> 

Example  No.  2. 
Equation,        2x -\- "^x -\-  %  =  5x  —  2a?  +  18 
Transposing,  2x  -{-  '^x  —  5x  -\-  2x  =  18  —  Q 

Uniting,  2a?  =  12 

Dividing,  x  =  ^ 

Problems. 


Find  the  value  of  x. 

1.  a;  +  4=  12 

2.  x^2,x  =  8 

3.  5x-2=  23 

4.     ZX  -  X  =  4:4: 

5.  lx-^x  =  144. 


6.  3a;  +  2^-4  =  £c  +  16 

7.  5a;  -  7  =  3a?  +  5 

8.  lx-\-2x-  x  =  ?,x-\-Z5 

9.  5aj  —  4:X  —  3x  -{-  6x  =  44 
10.  6^-8  -2a;  =  3^ +5 


(a)  Find  the  sum  of  the  ten  results. 


PART   II.  219 

Geometry. 

172.  Quadrilaterals  that  are  not  Parallelograms. 

a  c 


Trapezoid       \  \     Trapezium 


1.  Two  of  the  sides  of  a  trapezoid  are  parallel  and  two  are 
not  parallel.  In  the  trapezoid  represented  above  the  side 
ac  is  parallel  to  the  side  . 

2.  No  two  of  the  bounding  lines  of  a  trapezium  are  parallel. 

3.  In  the  trapezoid  represented  above  no  one  of  the  angles 
is  a  right  angle.  Name  the  angles  that  are  greater  than 
right  angles ;  the  angles  that  are  less  than  right  angles. 

4.  Draw  a  trapezoid  two  of  whose  angles  are  right  angles. 

5.  Can  you  draw  a  trapezoid  having  one  and  only  one 
right  angle  ? 

6.  Draw  a  trapezium  one  of  whose  angles  is  a  right  angle. 

7.  Can  you  draw  a  trapezium  having  more  than  one  right 
angle  ? 

8.  Every  quadrilateral  may  be  di- 
vided into  two  triangles.  Eemember 
that  the  sum  of  the  angles  of  two 
triangles  is  equal  to  four  right  angles. 
Observe  that  the  sum  of  the  angles  of 
the  two  triangles  is  equal  to  the  sum 
of  the  angles  of  the  quadrilateral.  So 
the  sum  of  the  angles  of  a  quadrilateral 
is  equal  to  four  right  angles. 


220  COMPLETE    ARITHMETIC. 

173.  Miscellaneous  Review. 

1.  If  two  of  the  angles  of  a  trapezoid  are  right  angles  and 
the  third  is  an  angle  of  60°,  how  many  degrees  in  the  fourth 
angle  ?     Draw  such  a  trapezoid.* 

2.  If  the  sum  of  three  of  the  angles  of  a  trapezium  is 
298°,  how  many  degrees  in  the  fourth  angle  ?  Draw  such  a 
trapezium.*    • 

3.  If  one  of  the  angles  of  a  triangle  is  an  angle  of  80°, 
and  the  other  two  angles  are  equal,  how  many  degrees  in 
each  of  the  other  angles  ?     Draw  the  figure.* 

4.  If  one  of  the  angles  of  a  quadrilateral  is  a  right  angle, 
and  the  other  three  angles  are  equal,  what  kind  of  a  quad- 
rilateral is  the  figure  ? 

5.  One  of  the  angles  of  a  quadrilateral  is  a  degrees; 
another  is  h  degrees ;  the  third  is  c  degrees.  How  many 
degrees  iti  the  fourth  angle  ? 

6.  The  smallest  angle  of  a  triangle  is  x  degrees ;  another 
angle  is  2  aj  degrees,  and  the  third  is  3  a?  degrees : 

Then  x-\-1x-^'^x  =  180. 
Find  the  value  ^i  x\  of  2  a? ;  of  3  a?. 

7.  643,265,245,350.  Without  performing  the  division 
tell  whether  this  number  is  exactly  divisible  by  9  ;  by  5  ;  by 
10;  by  25;  by  50  ;  by  12|;  by  18  ;  by  6 ;  by  15  ;  by  30 ; 
by  90;  byl6|.t 

8.  A  number  is  made  up  of  the  following  prime  factors : 
2,  2,  3,  3,  5,  7, 11.  Is  the  number  exactly  divisible  by  18  ? 
by  26  ?  by  35  ?  by  77  ?  by  21  ?  by  30  ?  by  45  ?  by  8  ? 

*  It  is  not  expected  that  this  drawing  will  be  accurate  in  its  angular  measure- 
ment—simply an  approximation  to  accuracy,  to  aid  the  pupil  in  recognizing  the 
comparative  size  of  angles. 

t  A  careful  study  of  pages  211-215  inclusive  will  enable  the  pupil  to  make  the 
statements  called  for  with  little  hesitation. 


FEACTIONS. 

174.  A  fraction  may  be  expressed  by  two  numbers,  one 
of  them  being  written  above  and  the  other  below  a  short 
horizontal  line  ;  thus,  |,  H,  f^f- 

175.  The  number  above  the  line  is  the  numerator  of  the 
fraction ;  the  number  below  the  line,  the  denominator  of  the 
fraction. 

176.  Kinds  of  Fractions. 

1.  A  fraction  whose  numerator  is  less  than  its  denomina- 
tor is  a  proper  fraction. 

|,  |,  ||,  are  proper  fractions. 

2.  A  fraction  whose  numerator  is  equal  to  or  greater  than 
its  denominator  is  an  improper  fraction. 

I^  6,  _2^i,  are  improper  fractions. 

Note. — The  fraction  .7  is  a  proper  fraction.  2.7  may  be  regarded 
as  an  improper  fraction  or  as  a  mixed  number.  If  it  is  to  be  con- 
sidered an  improper  fraction  it  should  be  read,  27  tenths;  if  a  mixed 
number,  2  and  7  tenths. 


3.  Such  expressions  as  the  following  are  compound  frac- 
tions : 

3  of  6      2  nf  1      5  of    "^ 

4.  A  fraction  whose  numerator  or  denominator  is  itself  a 
fraction  or  a  mixed  number,  is  a  complex  fraction. 

2.     2       1 

T>  Tr->  :t->  are  complex  fractions. 

4  3^  2^'  ^ 

221 


222  COMPLETE    ARITHMETIC. 

Fractions. 

5.  Any  fraction  that  is  neither  compound  nor  complex  is  a 
simple  fraction. 

|,  If,  14,  are  simple  fractions. 

6.  A  fraction  whose  denominator  is  1  with  one  or  more 
zeros  annexed  to  it  is  a  decimal  fraction. 

y\,  .7,  .25,  ^y^,  are  decimal  fractions. 

Note  1. — The  denominator  of  a  decimal  fraction  maybe  expressed 
by  figures,  or  it  may  be  indicated  by  the  position  of  the  right-hand 
figure  of  its  numerator  with  reference  to  the  decimal  point.  When 
the  denominator  is  thus  indicated,  the  fraction  is  called  a  decimal, 
and  is  said  to  be  written  decimally. 

Note  2.^A11  fractions  that  are  not  decimal  are  called  common 
fractions.  A  decimal  fraction  when  not  "  written  decimally  "  (or 
thought  of  as  written  decimally)  is  usually  classed  as  a  common 
fraction. 

7.  A  complex  decimal  is  a  decimal  and  a  common  fraction 
combined  in  one  number. 

.7-|-,  .25^,  .056f,  are  complex  decimals. 

177.  There  are  three  aspects  in  which  fractions  should 
be  considered. 

I.    THE   FRACTIONAL   UNIT   ASPECT. 

The  numerator  tells  the  number  of  things  and  the  denomi- 
nator indicates  their  name.  In  the  fraction  ^  there  are  5 
things  (magnitudes)  called  sevenths.  In  the  fraction  |  there 
are  five  fractional  units,  each  of  which  is  one  eighth  of  some 
other  unit  called  the  unit  of  the  fraction. 

Note. — The  function  of  the  denominator  is  to  show  the  number  of 
parts  into  which  the  unit  of  the  fraction  is  divided  ;  the  function  of 
the  numerator,  to  show  the  number  of  parts  taken. 


PART   II.  223 

Fractions. 

11.    THE   DIVISION   ASPECT. 

The  numerator  of  a  fraction  is  a  dividend,  the  denomina- 
tor a  divisor,  and  the  fraction  itself  a  quotient;  thus,  in  the 
fraction  f,  the  dividend  is  5,  the  divisor  8,  and  the  quo- 
tient |. 

Note. — In  the  case  of  an  improper  fraction,  as  |,  it  may  be  more 
readily  seen  by  the  pupil  that  the  numerator  is  the  dividend,  the 
denominator  the  divisor,  and  the  fraction  (|  =  2)  the  quotient;  but 
the  division  relation  is  in  every  fraction,  whether  proper  or  improper, 
common  or  decimal,  simple  or  complex. 

III.    THE   RATIO   ASPECT.* 

The  numerator  of  a  fraction  is  an  antecedent,  the  denom- 
inator a  consequent,  and  the  fraction  itself  a  ratio;  thus,  in 
the  fraction  y^^,  7  is  the  antecedent,  10  the  consequent,  and 
■^  the  ratio. 

Note  1 — This  relation  may  be  more  readily  seen  by  the  pupil  in 
the  case  of  an  improper  fraction.  In  the  fraction  J^,  12  is  the  ante- 
cedent, 4  the  consequent,  ^,  or  3,  the  ratio. 

Note  2. — Every  integral  number  as  loell  as  every  fraction  is  a  ratio. 
The  number  8  is  the  ratio  of  a  magnitude  that  is  8  times  some  unit 
of  measurement  to  a  magnitude  that  is  1  time  the  same  unit  of 
measurement. 

178.  Eeduction  of  Fractions. 

1.  The  numerator  and  the  denominator  of  a  fraction  are 
its  terms. 

2.  A  fraction  is  said  to  be  in  its  lowest  terms  when  its 
numerator  and  denominator  are  integral  numbers  that  are 
prime  to  each  other. 

*  This  may  be  omitted  until  the  book  is  reviewed. 


224  COMPLETE   ARITHMETIC. 

Fractions. 

3.  Reduce  |-|-g-  to  its  lowest  terms. 

Operation.  Explanation. 

1  c\^^^  —  ^^  Dividing  each  term  of  \^%  by  10,  we  have 

^^200  ~~  20*         1  tenth  as  many  parts,  which  are  10  times  as 

^n.        A  large.    Dividing  each  term  of  \%  by  4,  we  have 

4)jTT  =  — .  1  fourth  as  many  parts,  which  are  4  times  as 

large.     Hence,  \^  =  |.    But  4  and  5  are  prime 

to  each  other,  and  the  fraction  is  in  its  lowest  terms. 

Rule. — Divide  each  term  of  the  fraction  hy  any  common  divisor 
except  1,  and  divide  each  term  of  the  fraction  thus  obtained  hy  any 
common  divisor  except  1,  and  so  continue  until  the  terms  are  prime  to 
each  other. 

Reduce  to  lowest  terms  : 


275 
'^  -^375 

520 

156 

^  ^  270 

^  ^  340 

*-  -^210 

(6)^ 

^  ^  180 

(7)  "^ 
^  ^  405 

(8)  ^^ 
^  -^204 

(9)  ^^* 

(10)  f 

(a)  Find  the  sum  of  the  ten  results.  J 

4.  Reduce  |  to  higher  terms  —  to  120ths. 

Operation.  Explanation. 

120  -^  8  =  15.  Ill  T^  there  are  15  times  as  many  parts 

as  there  are  in  |,  and  the  parts  are  1  fifteenth 
5  X  15  _  2^      as  large.     Hence,  /j^  =  |. 
8x15        120 

*  Divide  each  term  by  12$. 
t  Divide  each  term  by  J. 

t  If  the  pupil  has  not  had  sufficient  practice  in  addition  of  fractions  to  do  this 
the  finding  of  the  sum  may  be  omitted  until  the  book  is  reviewed. 


PART  II.  225 

Fractions. 

Eeduce  to  higher  terms  —  to  160ths. 


(1)1 

(2)+i 

(3)  A 

(4)ii 

(5)  A 

(6)1 

(7)  A 

(8)U 

(9)H 

(10)11 

(a)  Find  the  sum  of  the  ten  results. 

5.  Two  or  more  fractions  whose  denominators  are  the 
same,  are  said  to  have  a  common  denominator. 

6.  Two  or  more  fractions  that  do  not  have  a  common 
denominator  may  be  changed  to  equivalent  fractions  having 
a  common  denominator. 

Example. 
f  and  I  may  be  changed  to  12ths,  24ths,  or  36ths. 

3-TT       T-^T        "S^TB        T  =  "3"6 

7.  Two  or  more  fractions  that  do  not  have  a  common 
denominator  may  be  changed  to  equivalent  fractions  having 
their  least  common  denominator.  The  1.  c.  d.  of  two  or 
more  fractions  is  the  1.  c.  m.  of  the  given  denominators. 

Example. 

Change  |i,  ^,  and  |^  to  equivalent  fractions  having  their 
least  common  denominator. 

Operation. 
(1)  The  1.  c.  m.  of  30,  40,  and  60  is  120. 
(2)120.30  =  4     3^.^ 
(3)120-.40  =  3     l^^^ 
(4)  120 -.60  =  2     |.^ 


226  COMPLETE    ARITHMETIC. 

Fractions. 

Eeduce  to  equivalent  fractions  having  their  1.  c.  d. 

1.  1^  and  ^\.  6.  ^\,  |,  and  ^i. 

2.  U  and  il  7.  ^\,  ^.  and  ^. 

3.  II  and  i|.  8.  ^\,  f,  and  |f. 
4  ^V  and  ^.  9.  |f,  ^\,  and  ||. 
5.  A  and  ||.  10.  If  A.  and  |f. 

(a)  Find  the  sum  of  the  twenty-five  fractions.* 


179.  To  Add  Common  Fractions. 

Rule. — Reduce  the  fractions  if  necessary  to  equivalent  fractions 
having  a  common  denominator,  add  their  numerators,  and  write  their 
sum  over  the  common  denominator. 


Example. 

Add 

H.  U'  and  U. 

(1)  The  1.  c.  m. 

of  45,  30,  and  60  is  180. 

(2)  H  =  tVj- 

U  =  ill-      1*  =  \U- 

(3)  AV  +  iff 

+  HI  =  «!• 

id  the  sum  of — 

^%  and  -^\. 

6.  1^,  J,  and  ^. 

TT  and  tV 

7.  ii,  i,  and  T-V- 

il  and  ^V 

8.  il.  i,  and  If. 

tV  and  if 

9.  A.  1.  and  If. 

A  and  ^. 

10.  A,  i,  and  A- 

1. 

2. 
3. 

4. 
5. 

(a)  Find  the  sum  of  the  ten  sums.* 

(For  a  continuation  of  this  work,  see  page  231.) 

*  This  may  be  omitted  until  the  subject  of  fractions  is  reviewed. 


PAKT  II.  227 

Algebraic  Fractions. 

a    oc      n 

180.  The  expressions  -,  -,  — -,  are  algebraic  fractions. 

h    4:    cd 

The  above  expressions  are  read,  a  divided  by  5,  x  divided  by  4, 
6  divided  by  cd. 

181.  Eeduce  to  lowest  terms: 


ah       /     a  X  h     \ 
a^  ~  \a  X  d  X  cbj' 

4a  _  /2  X  2  X  a\ 
•    65"  V2  X  3x  &/' 

abc  _  /a  X  h  X  c\ 
'  bcd~  \b  X  c  xdj' 


ab  -i-  a 

b 

a^  -h  a~ 

-a' 

4a  H-  2 

2a 

6b  -^2  ~ 

~3b 

abc      fa  X  b  X  c\       abc  -^  be     a 
bed  -^  be     d 


Let  a  =  2,  b  =  S,  e  =  5,  and  d  =  7,  and  verify. 

Observe  that  to  reduce  a  fraction  to  its  lowest  terms  we  have  only  to 
strike  out  the  factors  that  are  common  to  its  numerator  and  denominator. 

a^'ir 

4.  -^.   What  factors  are  common  to  both  numerator  and 
a^e 

denominator?     Reduce  and  verify. 

2     1 

5.  -JL,  AVTiat  factors  are  common  to  both  numerator  and 

denominator  ?     Reduce  and  verify. 

a^_  xf__  4a  +  45  ^ 

•  a'W  ~  '  xY  ~  '  6e^M~ 

abc  .  ^    2ax  Zx  +  6y 

ax  4:a  X  Iz 


228  COMPLETE    ARITHMETIC. 

Algebraic  Fractions. 

182.  Keduce  to  higher  terms : 

1.  Change  -p-  to  a  fraction  whose  denominator  is  ahc. 

°       DC 

2a  X  a       2a^        Let  a  =  2,  h  =  ^,  and  c  =  5,  and  vei'ify  tlie 
T^Xa^abc    reduction. 

2.  Change  ^ —  to  a  fraction  whose  denominator  is  2ay\ 

3x  X  y        3xy         Give  any  values  you  please  to  a,  x,  and  y, 
Ony  yr  y~  2ay^    ^^^  verify  the  reduction. 

183.  Keduce  to  equivalent  fractions  having  a  common 
denominator: 

X  y  Since   the   common    denominator   must  be 

^  ^^     d^^     exactly  divisible  by  each  of  the  given  denomi- 
nators, it  must  contain  all  the  prime  factors  * 
found  in  either  of  the  given  denominators.     The  new  denominator 
must  therefore  be  axaxhxd  =  a^bd ;    a^bd  -i-  ab  =  ad ;    a^bd  -^ 
a^d  =  b. 

X  X  ad       adx  y    xh        hy 

ah  X  ad  ~  a^hd  d^d  xh~  d^bd 

Give  any  values  you  please  to  a,  h,  d,  x,  and  y,  and  verify. 

4  Z  The  common  denominator  must  contain  the 

ah^  hc^      factors  a,  b,  b,  c,  c.     Reduce  and  verify. 

^    xy        ^  yz        The  common  denominator  is  5a.     Reduce  and 

3.  -^  and  ^^        .. 

-     5  a     verify. 

*  Since  the  numerical  values  of  the  letters  are  unknown,  each  must  be  regarded 
as  prime  to  all  the  others.  The  prime  factors,  then,  in  the  first  denominator  are  a 
and  6;  in  the  second,  a,  a,  and  d. 


PART   II. 


229 


Geometry. 
184.  Quadrilaterals. 

1.  All  the  geometrical  figures  on 
this  page  are  quadrilaterals ;  that  is, 
each  has  four  sides. 

2.  The  first  four  figures  are  par- 
allelograms ;  that  is,  the  opposite 
sides  of  each  flgitre  are  parallel. 

3.  The  first  two  figures  are  rec- 
tangular; that  is,  their  angles  are 
right  angles. 

4.  The  first  and  third  are  equi- 
lateral ;  that  is,  the  sides  are  equal. 

5.  There  is  one  equilateral  rec- 
tangular parallelogram.  Which  is 
it? 

6.  There  is  one  equilateral  paral- 
lelogram that  is  not  rectangular. 
Which  is  it  ? 

7.  There  is  one  rectangular  par- 
allelogram that  is  not  equilateral. 
Which  is  it  ? 

8.  The  sum  of  the  angles  of  each 

figure  on  the  page  is  equal  to  

right  angles. 

9.  Tell  as  nearly  as  you  can  the 
size  of  each  angle  of  each  figure. 


a              b 

1 
c              d 

Square. 

a                    L 
2 

c                   d 

) 

7 

Oblong. 

r-j 

1 

/ 

Rhombus. 

/a 

4 

1 

/ 

Rhomboid. 

a                     b\ 

Trapezoid. 


Trapezium. 


230  COMPLETE    ARITHMETIC. 

135.  Miscellaneous  Review. 

1.  The  difference  of  two  numbers  is  374-^;  the  smaller 
number  is  243^-J.     What  is  the  larger  number  ? 

2.  The  difference  of  two  numbers  is  a;  the  smaller  num- 
ber is  h.     What  is  the  larger  number  ? 

3.  James  had  a  certain  number  of  dollars  and  John  had 
three  times  as  many ;  together  they  had  196  dollars.  How 
many  had  each  ?     (^  +  3a?  =  196.) 

4.  William  had  a  certain  number  of  marbles ;  Henry  had 
twice  as  many  as  William,  and  George  had  twice  as  many  as 
Henry;  together  they  had  161.  How  many  had  each? 
{x-{-2x-\-4.x=  161.) 

5.  Divide  140  dollars  between  two  men,  giving  to  one  man 
30  dollars  more  than  to  the  other.     (ic  +  ^+  30  -  140.) 

6.  By  what  integral  numbers  is  30  (2x3x5)  exactly 
divisible  besides  itself  and  1  ? 

7.  By  what  is  ahc  {a  x  h  x  c)  exactly  divisible  besides 
itself  and  1  ? 

(1)  How  many  times  is  a  contained  in  ahc  ? 

(2)  How  many  times  is  h  contained  in  ahc  ? 

(3)  How  many  times  is  c  contained  in  ahc  ? 

(4)  How  many  times  is  ah  contained  in  ahc  ? 

(5)  How  many  times  is  ac  contained  in  ahc  ? 

(6)  How  many  times  is  he  contained  in  ahc  ? 

Ohserve  that  a  numher  composed  of  three  different  prime 
factors  has  exact  integral  divisors. 

8.  Change  |  to  60ths.     Is  f  more  or  less  than  |^  ? 

9.  Change  |  to  lOOths.     Change  |  to  lOOths. 
10.  Change  ^  to  lOOths.     Change  |  to  lOOths. 


FEACTIONS. 

(Continued  from  page  226.) 

186.  To  Subtract  Common  Fractions. 

jiULE. — Reduce  the  fractions  if  necessary  to  equivalent 
fractions  having  a  common  denominator,  find  the  difference 
of  their  numerators,  and  write  it  over  the  common  denominator. 

Example. 
From  ^1  subtract  ^j. 
.       (1)  The  1.  c.  m.  of  25  and  35  is  175. 
(2) 
(3) 

Compare  the  following : 

77  175ths-35  175ths  =  42  175ths. 
77  apples  —  35  apples  =  42  apples. 

Find  the  difference  of — 


H  =  tVt- 

A  = 

xVr 

i%V  -  t\V 

_   43 

-  TT"g" 

1. 

f  and  ^. 

10. 

i  and  i. 

2. 

t  and  A. 

11. 

1  and  ^. 

3. 

A-  and  |. 

12. 

f  and  ^\. 

4. 

H  and  I 

13. 

1  and  ^V 

5. 

1  and  ^. 

14. 

i  and  ^V- 

6. 

landi. 

15. 

1  and  A- 

7. 

i  and  |.  . 

16. 

1  and  f 

8. 

i  and  i- 

17. 

A  and  i. 

9. 

i  and  f 

18. 

If  and  i 

(a)  Find  the  sum  of  the  eighteen  differences. 

23X 


232  COMPLETE    ARITHMETIC. 

Fractions. 
187.  To  Subtract  one  Mixed  Number  from  another  when 
THE  Fraction  in  the  Subtrahend  is  greater  than 
the  Fraction  in  the  Minuend. 

Example. 
From  58|  take  32f. 
Operation.  Explanation. 

58|-  =  58^V  ^4  ^^  greater  than  ^j,  therefore  we  take  1 

32-|  =  324|-       unit  from  the  8  units,  change  it  to  24ths,  and 
Difference  "25^      ^^^  it  to  the  9  24ths. 

2  units  from  7  (8  —  1)  units  =  5  units.     3  tens  from  5  tens  —  2 
tens. 

I.  Find  the  difference  of— 


1. 

24-1  and  16f. 

6. 

35|  and  2^. 

2. 

29f  and  15f 

7. 

28f  and  14f. 

3. 

46i|  and  18^ 

8. 

36/^  and  8^. 

4. 

52f  and  31|. 

9. 

651-  and  22|. 

5. 

47|  and  18f. 

10. 

341-  and  27^. 

(a)  Find  the  sum  of  the  ten  differences. 

II.  Reduce  to  simplest  form — 

1.  5J  +  3i-5|. 

2.  6|-3i  +  4f 

3.  2i-li  +  3f. 

4.  7|  +  3|-1|. 

5.  6^-3i+5f 

7.  6f-2t,-l}-lf|  +  2|. 

8.  5J  +  4|  +  2i  +  3|  +  3i. 

(b)  Find  tlie  sum  of  the  eight  results. 


PART   II.  233 

Fractions. 
188.  To  Multiply  a  Fraction  by  an  Integer. 

Multiply  ^V  by  6. 
Operation  No.  1.  Operation  No.  2. 

6  times  -^^^  are  ||  =  If.  6  times  ^-^^\=  1|. 

1.  Observe  that  by  the  first  operation  we  obtain  || ;  that  in  ||  thei'e 
are  6  times  as  many  parts  as  there  are  in  ^^^  and  that  the  parts  ai'e 
of  the  same  size  as  those  in  Z^. 

2.  Observe  that  by  the  second  operation  we  obtain  | ;  that  in  |  thei-e 
are  the  same  number  of  parts  as  there  are  in  ^''j,  and  that  the  parts 
are  6  times  as  great  as  those  in  /j. 

Note. — The  7  of  ^^  may  be  regarded  as  a  dividend ;  the  24,  as  a 
divisor,  and  /j  itself  as  a  quotient.  In  ||,  we  have  a  dividend  6 
times  as  great  as  that  in  g'j,  the  divisor  remaining  unchanged.  In 
I  we  have  a  divisor  1  sixth  as  great  as  that  in  j\,  the  dividend 
remaining  unchanged.  Multiplying  the  dividend  or  dividing  the 
divisor  by  any  number  multiplies  the  quotient  by  the  same  number. 

Rule. — To  multvply  a  fraction  hy  an  integer,  multiply  its 
numerator  or  divide  its  denominator  hy  the  integer. 

I.  Find  the  product. 


1. 

A  X  4. 

5. 

f  x8. 

9. 

Ax  4. 

2. 

A  X  6. 

6. 

*x9. 

10. 

tV  X  6. 

3. 

iix5. 

7. 

f  X  8. 

11. 

A  X  5. 

4. 

Wx7. 

8. 

|x9. 

12. 

1  3  V  7 

30    X    /. 

(a)  Find  the  sum  of  the  twelve  products. 
II.  Find  the  product. 


1. 

H 

x7. 

5. 

4f  X  5. 

9. 

6.3   X  5. 

2. 

H 

x6. 

6. 

I|x4. 

10. 

m  X  4. 

3. 

7    7 
'^0 

x4. 

7. 

5|x5. 

11. 

4f    x6. 

4. 

3.7 

x5. 

8. 

8^x4. 

12. 

6^    x7. 

(b)  Find  the  sum  of  the  twelve  products. 


234  COMPLETE    ARITHMETIC. 

Fractions. 
189.   To  Divide  a  Fraction  by  an  Integer. 

Divide  f  by  3. 
Operation  No.  1.  Operation  No.  2. 

One  third  of  f  =  f .  One  third  of  i  =  ^V 

One  third  of  f  =  ^^  =  2. 

1.  Observe  that  by  the  first  operation  we  obtain  f  ;  that  in  |  there 
are  1  third  as  many  parts  as  there  are  in  f ,  and  that  the  parts  are  of 
the  same  size  as  those  in  |. 

2.  Observe  that  by  the  second  operation  we  obtain  /y ;  that  in  /^ 
there  are  the  same  number  of  parts  as  there  are  in  |,  and  that  the 
parts  are  1  third  as  great  as  those  in  f . 

Note  1. — The  6  of  |  may  be  regarded  as  a  dividend;  the  7  as  a 
divisor,  and  the  f  itself  as  a  quotient.  In  f  we  have  a  dividend  1 
third  as  great  as  that  in  |,  the  divisor  remaining  unchanged.  In 
g\  we  have  a  divisor  3  times  as  great  as  that  in  |,  the  dividend 
remaining  unchanged.  Dividing  the  dividend  or  multiplying  the 
divisor  by  any  number  divides  the  quotient  by  the  same  number. 

KuLE. — To  divide  a  fraction  by  an  integer,  divide  its 
•numerator  or  multiply  its  denominator  by  the  integer. 

I.    Find  the  quotient.       (See  p.  245,  problems  15  and  16.) 


1. 

A-4. 

4. 

1-^4. 

7. 

H^4. 

2. 

tV-5. 

5. 

1-^5. 

8. 

il-5. 

3. 

tV-20. 

6. 

1  -  20. 

9. 

il-2( 

(a)  Find  the  sum  of  the  nine  quotients. 

II.    Find  the  quotient.       (See  p.  245,  problems  17  and  18.) 


1. 

17i-^3. 

4.  18^^  -^  3. 

7. 

l^  -^  3. 

2. 

17|  -H  4. 

5.  18^3,  ^  4. 

8. 

16|-  -^  4. 

3. 

17i  -^  6. 

6.  18^^  -^  ^• 

9. 

16^  H-  6. 

(b)  Find  the  sum  of  the  nine  quotients. 


PART    II.  235 

Fractions. 

190.  To  Multiply  by  a  Fraction. 

$6  multiplied  by  3,  means,  take  3  times  $6.     $6  x  3  =  $18. 

$6  multiplied  by  2,  means,  take  2  times  $6.     $6  x  2  =  $12. 

$6  multiplied  by  2^,  means,  take  2  J  times  $6  ;  or  2  times  $6 

+  |-of  $6.     $6x2|-  =  $15. 
$6  multiplied  by  |-,  means,  take  ^  of  $6.     $6  x  |  =  $3. 
$6  multiplied  by  |,  means,  take  |  of  $6.     $6  x  f  =  $4. 

To  THE  Teacher. — Require  the  pupil  to  examine  the  preceding 
statements  until  he  clearly  understands  that  to  multiply  by  a  fraction 
is  to  take  such  part  of  the  multiplicand  as  is  indicated  by  the  frac- 
tion. Thus :  to  multiply  48  by  |  is  to  take  three  fourths  of  48 ;  that 
is,  three  times  1  fourth  of  4^.  It  will  thus  be  clear  that  multiplication 
by  a  fraction  involves  both  multiplication  and  division. 

Example  I. 

Multiply  24  by  |. 
1  fourth  of  24  is  6. 
3  fourths  of  24  are  18. 

Example  III. 
Multiply  2 75 f  by  f. 
1  fourth  of  275|  is  68395-. 
3  fourths  of  275f  are  206yV 

'  EuLE. — To  multijply  hy  a  fraction,  divide  the  multiplicand 
hy  the  denominator  of  the  fraction  and  midtiply  the  quotient 
thus  obtained  by  the  mtmerator  of  the  fraction. 

Observe  that  in  practice  we  may,  if  more  convenient,  multiply  the 
multiplicand  by  the  numerator  of  the  fraction,  and  divide  the  prod- 
uct thus  obtained  by  the  denominator.  To  multiply  12  by  |  we  may 
take  3  times  1  fourth  of  12  or  1  fourth  of  3  t'mes  12,  as  we  choose. 


Example 

II- 

Multiply  f  by  |. 
1  fourth  of  f  is  ^^. 

3  f^  .rths  of  1 

are  /^. 

Example 

IV. 

Multiply  346f  by  2^. 
Two  times  346f  =  6924. 

1  half  of  346f 

=  1731. 

6924  +  173|z 

=  866  Ans. 

236  COMPLETE    ARITHMETIC. 

Fractions. 

I.  Find  the  product.       (See  p.  245,  problems  19  and  20.) 

1.  345  X  f  4.  263  x  f  7.  263  x  f 

2.  345  X  yV*       5.  263  x  |.  8.  576  x  |. 

3.  345  X  i  6.  263  x  f  9.  576  x  |. 

(a)  Find  the  sum  of  the  nine  products. 

II.  Find  the  product.       (See  p.  245,  problems  21  and  22.) 

I'-Xvi  4.5^1  75vl 

2-  t\  X  A-t         5.  f  X  |.  8.  J  X  |. 

3.  -r\  X  i.  6.  I  X  |.  9.  i  X  I- 

(b)  Find  the  sum  of  the  nine  products. 

III.  Find  the  product.       (See  p.  246,  problems  23  and  24.) 

1.  372^  X  f    4.  523f  x  f  7.  523f  x  i- 

2.  372J^  X  yV   5.  523f  x  f.  8.  153i-  x  f. 

3.  372^  X  i-    6.  523f  x  f  9.  153|-  x  f. 

(c)  Find  the  sum  of  the  nine  products. 

IV.  Find  the  product.       (See  p.  246,  problems  25  and  26.) 


1. 

462f  X  2^. 

6. 

346-J  X  3|. 

2. 

462|  X  3yV 

7. 

346J  X  2^. 

3. 

462f  X  2f 

8. 

2751-  X  4f 

4. 

346 J  X  2i. 

9. 

2751-  X  3^. 

5. 

346J  X  3f. 

10. 

275J  X  2^. 

(d)  Find  the  sum  of  the  ten  products. 


*  Take  3  times  1  tenth  of  345,  or  1  tenth  of  3  times  345. 

t  Lead  the  pupil  to  see  that  in  problems  of  this  kind  the  correct  result  may  be 
obtained  by  "  miiUiplyuig  the  mimerators  together  for  a  new  numerator  and  the  denomina- 
tors together  for  a  new  denominator";  that  in  so  doing  he  divides  the  multiplicand  by 
the  denominator  of  the  multiplier  and  multiplies  the  quotient  so  obtained  by  the 
numerator  of  the  multiplier. 


PART    II. 
Algebraic  Fractions. 


237 


a      c 


1.  c.  d.  =  hd 


bd^h  =  d     -^-^^ 


a  X  d 
b  X  d 


ad 
bd 


bd 


G  X  b 
d  X  b 


be 
bd 


ad       be      ad  -\-  be 


bd^  bd~ 


bd 


I  +  -.     Led.  =  21 


21  -=-3  =  7. 


2x7 
3x7 


14 
2l 


21 
14 


2l"^21 


7  =  3. 
15      29 


5  X  3  _15 
7x3~2r 


21 


Observe  that  in  cases  like  the  above,  in  which  the  denominators 
are  prime  to  each  other,  the  1.  c.  d.  is  the  product  of  the  given 
denominators,  and  each  new  numerator  may  be  found  by  multiplying 
the  given  numerator  by  the  denominator  of  the  other  fraction. 


191.  Problems  in  Addition  and  Subtraction. 


a      b      ay       bx      ay  -f  bx 
X      y     xy      xy  xy 

Let  a  =  2,  6  =  3,  ic  =  5,  ?/  =  7,  and  verify. 
a      b      ay      bx      ay  —  bx 


2.-- 

X      y     xy      xy 


xy 


Assign  a  numerical  value  to  each  letter  and  verify. 


3.  --- 

X      y 


4. 


Solve. — Then  let  £c  =  5  and  y  =  1  and 
verify. 


X      y 


238 


COMPLETE   ARITHMETIC. 


Algebraic  Fractions. 

192.  Problems  in  Multiplication  and  Division. 

Example  I. 


a             ac 

I'-'l-n 

L( 

*.                   Ct      -L             r-               J                   0X1-^C2X36., 

3t  a  -  2,  &  -  5,  and  c  =  3  ;  then       =            .=     =  1\. 

ODD 

Example  II. 

a              a 

2      .       2 

b^'-be 

5^^~15 

L( 

it  a  =  2,h  =  5,  and  c  =  S  :  then — =  — 

he      5x3      15 

Example  III. 

a      c      ac 

2     3       6         • 

h^d^M 

5  ^  7  ~  35 

L( 

3t  a  -  2,  5  -  5,  c  -  3,  and  6^  -  7 ;  then  ^^2x3^  6 

od      5x7      35 

I.  Find  the  product  and  verify  as  above. 

1. 

a^                             ^    X                              ^    ax      ^ 
V3  X  c                      3.       xy                     5.  — -  X  3a 
¥                                  y^     "^                            bx 

2. 

-^.X  X                     4.        xa                    6.  -x5 
y'                                  cd                                7 

II.  Find  the  quotient  and  verify  as  above. 

1. 

a'^                             ^    X                              ^    ax      ^ 
jj^c                       S,  ^^y                      5.         ^  3a 
b^                                  y^     *^                            hx 

2. 

X                               .    ah 
—  ■i-x                      4.  — 
y'                                 cd 

o     X        ^ 

-5-  a                    6.  -  -^  5 

PART   II. 


239 


Qeometry. 
193.  The  Protractor. 


Measure 
angle  c. 


240  COMPLETE   ARITHMETIC. 

194.  Miscellaneous  Review. 

1.  A  piece  of  land  in  the  form  of  an  equilateral  triangle 
measures  on  one  side  46^^^  rods.  What  is  the  distance 
around  it  ? 

2.  The  perimeter  of  a  piece  of  land  that  is  an  exact  square 
is  246|-  feet.     How  far  across  on  one  side  ? 

3.  The  length  of  a  certain  rectangular  field  is  three  times 
its  breadth ;  its  perimeter  is  360  rd.  What  is  its  breadth  ? 
Its  length  ? 

4.  If  f  of  the  value  of  a  farm  is  $2154,  what  is  -|  of  the 
value  of  the  farm  ? 

Note. — If  f   of  a  certain  number  is  24,  what  is  the  number? 
What  is  I  of  the  number  ? 

5.  I  spent  f  of  my  money  and  had  $3.60  remaining, 
(a)  How  much  did  I  spend  ?  (b)  What  I  had  remaining, 
equals  what  part  of  what  I  spent  ? 

6.  Change  ^^^  to  an  equivalent  fraction  whose  denomi- 
nator is  30. 

7.  Change  f  to  an  equivalent  fraction  whose  numerator 
is  30. 

a' 

8.  Change  —  to  an  equivalent  fraction  whose   deuomi- 

nator  is  he. 

9.  Multiply  I  by  f  and  multiply  the  product  by  25. 

10.  Multiply  1^  by  ^|  and  multiply  the  product  by  25. 

11.  If  -I  of  an  acre  of  land  is  worth  $36,  how  much  are 
37^  acres  worth  at  the  same  rate  ? 

12.  The  rent  of  a  house  for  2  yr.  4  mo.  was  $840.    What 
was  the  rate  per  year  ? 


FRACTIONS. 


395.   To  Divide  by  a  Fraction. 
Example  I. 

Divide  6  by  |. 
Operation  No.  1.  Operation  No.  2. 

1^2    —   34- 

^       H  —  2-T 


6  =  V-. 


_  Q  * 


6  -f- 1  =  6  times  f  =  -i/  =  9. 


Operation  No.  1. 


7   —   35 

S  -  TO- 


Example  II. 
Divide  ^  by  f . 


Operation  No  2. 


3    —   24 

r  -  TIT- 


35^24 
TO"    •    T¥ 


111   + 
^^4'  + 


1  -  f  =  l§ 

J  -  f  =  I  of  I  =  If  =  m. 


From  the  foregoing  operations  the  following  rules  for  dividing  by 
a  fraction  are  obtained  : 

Rule  I. — Reduce  the  dividend  and  the  divisor  to  like  frac- 
tional units,  then  divide  the  numerator  of  the  dividend  hy  the 
numerator  of  the  divisor. 

Rule  II. — ''Invert  the  divisor  and  proceed  as  in  multipli- 
cation" 

Observe  that  the  inverted  divisor  shows  the  number  of  times  the 
divisor  is  contained  in  1 :  then  in  6  it  is  contained  6  times  as  many 
times ;  in  4,  4  times  as  many ;  in  |,  |  as  many ;  in  |,  |  as  many,  etc. 

*  18  thirds  +  2  thirds  =  9.  $  35  fortieths  -+-  24  fortieths  =  IJl. 

1 1  =  g.    3  thirds  +  2  thirds  =  IJ  =  e.         g  l  =  g.    5  fifths  +  3  fifths  =  1|  =  g. 

241 


242  COMPLETE   ARITHMETIC. 

Fractions. 

I.    Find  the  qUOtiert.       ( see  page  246,  problems  27  and  28.) 


1.  46  -^  I 

4.  375  ^  f 

7.  196  ^f 

2.  46  ^  f . 

5.  375  H- 1. 

8.  196-^3. 

3.  46-^1. 

6.  375  H-  f. 

9.  196-^  |. 

(a)  Find  the  sum  of  the  nine  quotients. 

II.    Find  the  quotient.       (See  page  246,  problems  29  and  30.) 


1. 

|-|. 

4. 

A-*- 

7. 

H-l 

2. 

1-i 

5. 

A-i 

8. 

4i-^l- 

3. 

l-i*- 

6. 

A-lf 

9. 

^T  "^  T?- 

(b)  Find  the  sum  of  the  nine  quotients. 

III.    Find  the  quotient.       (See  page  246,  problems  31  and  32.) 


1.  5i*|. 

4. 

24|^|. 

7. 

19|-t- 

2.  5J^|. 

5. 

24|^|. 

8. 

19|^f 

3.  5i^|. 

6. 

24i^|. 

9. 

19|  -  f. 

(c)  Find  the  sum  of  the  nine  quotients. 

IV.    Find  the  quotient.       (See  page  246,  problem  33.) 


1. 

325|--^2f 

6. 

174^-^-2  J. 

2. 

325|--f-2f. 

7. 

174^ -^2f 

3. 

325^^31 

8. 

174^-4-31. 

4. 

325^-^7  J. 

9. 

1741- -^  7  J. 

5. 

3251- H-H. 

10. 

174^^  If 

(d)  Find  the  sum  of  the  ten  quotients. 


PART   II.  -  243 

Fractions. 

196.  To  Eeduce  Complex  Fractions  to  Simple 
Fractions. 

I  is  read,  |  divided  by  |.     f  ^  7  ^  3  of  s  =  2  4  ^  e . 

T.  Eeduce  to  their  simplest  forms. 

5555 

^  T  T  6 

(a)  Find  the  sum  of  the  four  fractions. 

Observe  that  a  complex  fraction  may  be  reduced  to  a  simple 
fraction  by  multiplying  its  numerator  and  denominator  by  some 
number  that  will  in  each  case  give  an  integral  product.  When  this 
number  can  be  easily  discovered  by  inspection  this  is  a  convenient 

method  of  reduction :  thus  i  =  ^      9  ~  Tl' 

II.  Eeduce  to  their  simplest  forms.     (See  page  246,  problem  34.) 

6  1  K  11 

1.  g  Z.  ^  6.  ^  4.    2 

(b)  Find  the  sum  of  the  four  fractions. 

III.  Eeduce  to  their  simplest  forms. 

.    8  ^8  ^8  8 

1-T  2.^  3.^  4.- 

^  T  ¥  "5 

(c)  Find  the  sum  of  the  four  fractions. 

IV.  Eeduce  to  their  simplest  form. 

1    l!i*       2   IM         3    ^         4   ^ 
100  100  100  100 

♦Divide  the  nxunerator  and  denominator  by  12J. 


244  COMPLETE    ARITHMETIC. 

Fractions. 
197.   Pkactical  Application  of  the  Preceding  Eules. 

Page  224,  problems  1  and  2. 

1.  B  owned  a  farm  of  375  acres;  he  gave  to  his  son 
275  acres.     What  part  of  the  farm  did  the  son  receive  ? 

2.  Mr.  L.  earned  $650  in  one  year;  of  this  sum  he  ex- 
pended S520.     What  part  of  his  earnings  did  he  expend? 

Page  225,  problems  1  and  2. 

3.  A  lady  owned  ^  of  an  acre  of  land.  How  many 
160ths  of  an  acre  did  she  own? 

4.  Benton  walked  ^^  of  a  mile.  Express  the  distance  he 
walked  in  160ths  of  a  mile. 

Page  226,  Art.  179,  problems  1  and  6. 

5.  In  a  certain  furnace  -^^  of  a  ton  of  coal  was  consumed 
in  one  day,  and  -^-^  of  a  ton  the  next  day.  What  part  of  a 
ton  was  consumed  in  the  two  days  ? 

6.  Mr.  Luker  has  three  lots  of  land ;  in  the  first  lot  there 
are  |-J  of  an  acre ;  in  the  second,  i  of  an  acre,  and  in  the 
third,  l^f  of  an  acre.     How  many  acres  in  all  ? 

Page  231,  Art.  186,  problems  3  and  4. 

7.  Of  -^^  of  a  mile  of  board  fence  i  of  a  mile  was  burned. 
What  part  of  a  mile  remained  ? 

8.  Mr.  Keynolds  had  put  into  the  bank  ^-J-  of  his  annual 
salary;  he  drew  from  this  money  a  sum  equal  to  -J  of  his 
salary.     What  part  of  his  salary  remained  in  the  bank  ^ 

Page  232,  Art.  187,  L,  problems  4  and  5. 

9.  From  52|-  tons  of  hay,  were  sold  and  delivered  31 1 
tons.     How  many  tons  remained  of  the  unsold  hay  ? 

10.  On  Monday  James  rode  47f  mi. ;  on  Tuesday,  18|  mi. 
How  much  further  did  he  ride  Monday  than  Tuesday  ? 


PART   II.  245 

Page  233,  Art.  188,  I.,  problems  1  and  5. 

11.  If  a  street  car  makes  a  round  trip  in  -^  of  an  hour,  in 
how  long  a  time  can  it  make  4  such  trips  ? 

12.  If  J  yd.  of  ribbon  are  required  to  trim  a  hat,  how 
much  ribbon  will  be  required  to  trim  8  such  hats  ? 

Page  233,  Art.  188,  II.,  problems  1  and  5. 

13.  At  3^  dollars  a  cord,  what  is  the  cost  of  7  cords  of 
wood  ? 

14.  If  Henry  rides  his  wheel  at  the  rate  of  4|  miles  an 
hour,  how  far  does  he  ride  in  5  hours  ? 

Page  234,  Art.  189,  I.,  problems  1  and  4. 

15.  If  yV  of  a  yd.  of  ribbon  is  cut  into  4  equal  pieces, 
what  part  of  a  yard  is  each  piece  ? 

16.  John  hoes  4  rows  of  corn  in  |  of  an  hour.  In  what 
part  of  an  hour  does  he  hoe  1  row  ? 

Page  234,  Art.  189,  II.,  problems  1  and  5. 

17.  A  horse  traveled  17^  miles  in  3  hours.  What  was 
his  rate  per  hour  ? 

18.  A  farmer  divided  a  field  containing  18y3-g-  acres  into  4 
equal  lots.     How  many  acres  in  each  lot  ? 

Page  236,  I.,  problems  1  and  5. 

19.  At  $345  an  acre,  what  is  the  cost  of  |^  acre  ? 

20.  At  $263  an  acre,  what  is  the  cost  of  f  acre  ? 

Page  236,  II.,  problems  1  and  9. 

21.  A  piece  of  land  in  the  form  of  a  rectangle  is  -{'^  of  a 
mile  long  and  |^  of  a  mile  wide.  The  piece  is  what  part  of 
a  square  mile  ? 

22.  At  i  a  dollar  per  yard,  what  is  the  cost  of  |-  of  a  yard 
of  silk  ? 


246  COMPLETE    ARITHMETIC. 

Page  236,  III.,  problems  1  and  9. 

23.  A  strip*  of  land  372|-  rods  long  and  ^  a  rod  wide  con- 
tains how  many  square  rods  ? 

24.  At  $153-|-  an  acre,  find  the  cost  of  -|-  of  an  acre. 

Page  236,  IV.,  problems  1  and  8. 

25.  At  $46 2 1  a  mile,  what  is  the  cost  of  grading  2^^.  miles 
of  road? 

26.  How  many  square  feet  in  a  piece  of  land  275^^  ft.  by 
4J  ft.? 

Page  242,  I.,  problems  1  and  5. 

27.  At  |-  a  dollar  a  bushel,  how  many  bushels  of  potatoes 
can  be  bought  for  46  dollars  ? 

28.  At  |-  of  a  dollar  a  bushel,  how  many  bushels  of 
apples  can  be  bought  for  375  dollars? 

Page  242,  II.,  problems  1  and  5. 

29.  At  I  of  a  dollar  a  yard,  how  many  yards  of  cloth  can 
be  bought  for  -J  of  a  dollar  ? 

30.  At  I  of  a  dollar  a  yard,  what  part  of  a  yard  of  cloth 
can  be  bought  for  -^\  of  a  dollar  ? 

Page  242,  III.,  problems  1  and  6. 

31.  If  a  rectangular  diagram  on  the  blackboard  contains 
5^  square  feet  and  is  f  of  a  foot  wide,  how  long  is  it? 
Make  the  diagram. 

32.  At  Sf  a  bushel,  how  many  bushels  of  meal  can  be 

bought  for  S24|  ? 

Page  242,  IV.,  problem  1. 

33.  A  strip  of  land  contains  325^  square  rods,  and  is  2| 
rods  wide.     How  long  is  it  ? 

Page  243,  II.,  problem  1. 

34 .  Five-sixths  of  an  hour  is  what  part  of  5  hours  ? 


PART    II.  247 

Algebraic  Fractions. 

198.  Problems  in  Division  with  a  Fraction  for  a 
Divisor. 

Example  I. 
See  page  241,  Rule  11,  and  Observation. 

Let  a  =  1,1  -  2,  and  c  =  3 ;  then-—  =  — - —  =  -7r=  104 

&  2  2  ^ 


Example  TI. 
See  page  241,  Rule  II,  and  Observation. 


a      b      ac 
X  '  c  ~  bx 


10  •  3  "  20      ^^ 


Let  a  =  7,05  =  10,6  =  2,  andc  =  3;  then-— =  - ^t^ttt^^I 

bx     2  X 10     20 


YH 


I.  Find  the  quotient  and  verify  as  above. 


1.  x-^ 


2.2/- 


X 


4.  b 


y 


5.  xy  ^  - 
a 


6.  yz 


II.  Find  the  quotient  and  verify  as  above. 


.     a      b 

1.  -  -^  — 

X     y 

6.  -  -i-  — 

y      z 

^    b      X 

0.    —  -i-  — 

c      y 

b^c_ 
X  '  y 


4. 


a      b 


z       X 


6. 


c       X 
d'"  y 


248 


COMPLETE    ARITHMETIC. 

Algebraic  Fractions. 

199.  Miscellaneous  Exercises. 

Example  I. 


h      ac  A-h 
c  c 


5       5  5 


Let  a  =  3,b  =  2,  and  c  =  5. 
c  5  5  5 


I.  Keduce  to  improper  fractions  and  verify  as  above. 


1.   X  + 


y 


2-^+^ 


3.^-1-1 


Example  II. 


ab  4-  c  c 


11  _  „2 


G  5        2 

Let  a  r=  2,  &  =z  3,  and  c  :i-  5;  then  a+-=  2  +-:=  3- 

0  So 


II.  Keduce  to  mixed  numbers  and  verify. 


1. 


ax  -{-b 


2. 


bij  +  c 


s.'l+I 


III.  Reduce  and  verify. 


2    — 
^-    3 


4.  A 
2 


PART  II.  249 

Geometry. 
200.    CONSTRUCTIOX   PROBLEMS — TRIANGLES. 


1.  Draw  a  triangle.  Make  the  side  ah  3  inches  long. 
Make  the  angle  a,  45°.  Make  the  angle  h,  45°.  Prove 
your  work  by  measuring  the  angle  c  which  should  be  an 
angle  of  degrees. 

Observe  that  if  two  angles  of  any  triangle  and  the  length 
of  the  included  side  are  given,  the  triangle  may  he  drawn. 

2.  Draw  a  triangle  making  one  of  the  angles  40°,  another 
60°,  and  the  included  side  5  inches  long.  The  third  angle 
should  measure  degrees.  Prove  your  work  by  measur- 
ing the  third  angle. 

Measure  the  sides  carefully  and  observe  that  the  longest  side 
is  opposite  the  largest  angle,  and  the  shortest  side  opposite  the 
smallest  angle. 

3.  Draw  several  triangles  of  different  shapes  and  sizes. 
Convince  yourself  by  measurement  with  the  protractor  that 
the  sum  of  the  three  angles  of  any  triangle  is  degrees. 

4.  Draw  several  triangles  of  different  shapes  and  sizes. 
Convince  yourself  by  measurement  with  a  ruler  that  the  sunt 
of  two  sides  of  any  triangle  is  greater  than  the  third  side  of 
the  same  triangle. 

5.  Attempt  to  draw  a  triangle  whose  sides  are  6  inches, 
3|^  inches,  and  2^  inches. 


250  COMPLETE    ARITHMETIC. 

201.    Miscellaneous  Review. 

1.  If  |-  of  a  cord  of  wood  cost  $4.50  how  much  will  27 
cords  cost  at  the  same  rate  ? 

2.  If  2^  tons  of  coal  cost  $12.60,  how  much  will  17|-  tons 
cost  at  the  same  rate  ? 

3.  A  man  owned  Yy^^-  acres  of  land;  he  sold  2 J  acres. 
What  fractional  part  of  his  land  did  he  sell  ? 

4.  The  sum  of  two  fractions  is  1  -^^ ;  one  of  the  fractions 
is  ^ .     What  is  the  other  fraction  ? 

5.  The  product  of  two  fractions  is  ^f ;  one  of  the  frac- 
tions is  f .     What  is  the  other  fraction  ? 

6.  If  a  furnace  consumes  y^^  of  a  ton  of  coal  a  day,  in 
how  many  days  will  5^  tons  be  consumed  ? 

7.  How  many  pounds  of  sugar  at  4(f  a  pound  must  be 
given  for  27|-  pounds  of  butter  at  23(f  a  pound? 

8.  How  many  pounds  of  coffee  at  S3^(f  a  pound  must  be 
given  for  15 J  dozen  eggs  at  20^  a  dozen? 

9.  Which  is  the  greater  fraction,  |  or  |  ? 

10.  Multiplying  both  terms  of  a  fraction  by  the  same 
number  does  not  change  the  value  of  the  fraction.  Does 
adding  the  same  number  to  both  terms  of  a  fraction  change 
the  value  ? 

11.  Dividing  both  terms  of  a  fraction  by  the  same  num- 
ber does  not  change  the  value  of  the  fraction.  Does  sub- 
tracting the  same  number  from  both  terms  of  a  fraction 
change  the  value  ? 

12.  Change  :|^  to  a  fraction  whose  denominator  is  46. 

13.  Change  ;^  to  a  fraction  whose  denominator  is  15. 


FEACTIONS. 

202.   To   Change   Decimals   to   Common  Fractions  and 
Common  Fractions  to  Decimals. 

Example  I. 

Change  .36  to  a  common  fraction  in  its  lowest  terms. 

36         9 

.36= =  — 

100      25 

Eeduce  to  common  fractions.  - 

1.  .45  3.  .375  5.  .55  7.  .625 

2.  .045  4.  .0375  6.  .055  8.  .0625 

(a)  Find  the  sum  of  the  eight  decimals. 

(b)  Find  the  sum  of  the  eight  common  fractions. 

Example  II. 

Change  |  to  a  decimal. 

Operation.  Explanation. 

8)3.000  3  over  8  means    3   divided   by   8.     We   therefore 

r;73       annex  zeros  to  the  numerator  3,  and  perform   the 
.o  I  O        , .    .  . 

division. 

One  eighth  of  30  tenths  is  3  tenths  with  a  remainder  of  6  tenths,  etc. 


Eeduce  to  decimals. 

1. 

2. 

i 

3. 
4. 

5-  i 

6-  fl 

7. 
8. 

1 

(c) 

Find  the  sum 

of  the  eight 

decimals. 

251 


252  COMPLETE    ARITHMETIC. 

Fractions. 

Example  III. 

Change  f  to  a  decimal. 

Operation.  Explanation. 

7)2.00  It  will  be  observed  that,  however  far  this 

90^1  division  may  be  carried,  there  is  always  a  re- 

mainder.    The  fact  that  there  is  a  remainder 
7)2.000  is  indicated  by  writing  the  plus  sign  after  the 

9854-  ^^^^  figure  of  the  decimal.     The  first  quotient 

may  be  read  28  hundredths,  plus. 
7)2.0000  Observe,  too,  that   the   error   in   the   first 

.2857+  answer  is  less  than    1   hundredth,  since   the 

true  quotient  is  more  than  28  hundredths  and 
less  than  29  hundredths.  AVe  may  therefore  say  that  the  first  result 
is  true  to  hundredths ;  the  second,  true  to  thousandths. 


Reduce  to  decimals,  true  to  thousandths. 

1. 4 

4-tV 

7.   J 

10.  1 

2.  f 

5.    ,3^ 

8.  1 

11.  f 

3.  f 

6-tV 

9.  i 

12.  i 

(a)  Find  the  sum  of  the  twelve  common  fractions. 

(b)  Find  the  sum  of  the  twelve  decimals.* 

Determine  which  of  the  following  fractions  can  be  redu(ied 
to  terminating  decimals  f  and  which  cannot. 


1. 

f 

3. 

1 

5. 

1 

7.  f 

9. 

A 

2. 

T 

4. 

\ 

6. 

t 

8.  A 

10. 

tV 

Observe  that  if  a  fraction  is  in  its  lowest  terms  and  the  denom- 
inator contains  any  other  prime  factor  besides  2's  and  o's,  the  fraction 
cannot  be  reduced  to  a  "terminating"  decimal.     Can  you  tell  why? 

*  Observe  that  the  difference  between  a  and  6  must  be  less  than  12  thousandths. 
Why? 

t  See  foot-note,  page  253. 


PART   II.  253 

Fractions. 

203.    To   Eeduce   a   Complex   Decimal*   to   a    Common 

Fraction. 

Example. 
Change  .27^^^  to  a  common  fraction. 

Operation.  Explanation. 

27^  Writing  the  denominator,  M-e  have 

•^ 'tt  =     -j^QQ  27t\  divided  by  100. 

273—300  27 r\  reduced  to  llths  =  \»A 

.3^(M)_  -5-  100  =  yWg-  =  -j-V  -T?  -  100  =  AVtt  =  rV 

Eeduce  to  common  fractions. 


1.  .38i 

4. 

.24^- 

7. 

.611 

2.  .83^ 

5. 

•35A 

8. 

.16| 

3.  .45f 

6. 

•40A 

9. 

.544 

(a)  Find  the  sum  of  the  nine  common  fractions. 

(b)  Find  the  sum  of  the  nine  complex  decimals. 

Note. — If.  the  division  be  carried  sufficiently  far  in  any  non-ter- 
minating decimal  there  will  be  found  a  certain  figure  or  set  of  figures 
that  is  constantly  repeated :  thus, we  may  have,  .3666666,  or  .27272727, 
ar  .5236236236.  The  part  repeated  is  called  a  repetend,  and  may  be 
«vTitten  thus:  .36,  .27,  .5236.  It  is  a  curious  fact  that  the  real 
denominator  of  any  repetend  is  as  many  9's  as  there  are  figures  in 
the  repetend  ;    .36  =  .3§,  .27  =  f|,  .5236  =  .5f||. 

*  Decimals  that  are  complete  without  the  annexation  of  a  common  fraction  are 
said  to  be  terminating  decimals.  .24  is  a  terminating  decimal.  .666666  +  is  a  non- 
terminating  decimal.  A  decimal  with  a  common  fraction  annexed,  as  .33J,  is  some- 
times called  a  complex  dednud. 


254  COMPLETE    ARITHMETIC. 

Fractions. 

204.  To  Keduce  a  Fraction  to  Hundredths. 

Examples. 

J      o  4)3.00  ^         p    100  75  ^r, 

1.3        ^;orfof-=_or.75. 

II.  I      3)2:00        rf  100^661^^ 

^  .661  ^       100     100  ^ 

-r-TT        7  8)7.00  „         n    100  874  o„, 

IV.  ^V    ^^^-^   ;or^Vofi^=-^or.02i. 
TO  ^2f        ^0         100      100  ^ 

I.  Eeduce  to  hundredths. 

1.    ^  6.  4-  11.    1  16.    3 


-L  71  122  175 


Ql  «3  IQll  1R9 

O.         g-  O.       g^  10.      -f-fy  10.      -j^ 

4.3  Q2  14.'"  1Q1 

5.    k  10.  I  15.  ^3^  20.  ^ 

(a)  Find  the  sum  of  the  twenty  decimals. 

(b)  Find  the  sum  of  the  twenty  common  fractions. 

II.  Eeduce  to  hundredths. 

1. 
2. 
3. 

(a)  Find  the  sum  of  the  nine  decimals. 

(b)  Find  the  sum  of  the  nine  common  fractions. 


tVV 

4-tV\ 

7-   T*,V 

1  5 

5.  AV 

8-tVt 

TT~S 

tVV 

6-  AV 

^■^ 

PART   II.  25o 

Fractions. 

III.  Reduce  to  hundredths. 

Note. — Such  fractions  as  the  following  may  be  easily  reduced  to 
hundredths  by  dividing  the  numerator  and  the  denominator  of  each 
by  that  number  which  will  change  the  denominator  to  100.* 

1. 

2. 
3. 
4. 

(a)  Find  the  sum  of  the  twelve  results. 

(b)  Find  the  sum  of  the  twelve  common  fractions. 

IV.  Reduce  to  hundredths. 

Note. — Multiply  the  numerator  and  the  denominator  of  each 
fraction  by  that  number  which  will  change  the  denominator  to  100. 

1. 


Ttnr 

5. 

45 
40  0 

9. 

AV 

AV 

6. 

4  0  0 

10. 

AV 

^ 

7. 

1  20 
TOD 

11. 

AV 

Ui 

8. 

m 

12. 

m 

7 
20 

-i 

7.i2 

20 

14 
25 

•^•fo 

8.  il 
25 

27 

50 

6.    ** 
12J 

9.^ 

50 

3. 

(a)  Find  the  sum  of  the  nine  decimals. 
V.  Reduce  to  hundredths. 

2.  t'A  5-  U  8-  AV 

Q      1  5  4  fi      16  9     JL  9  6. 

(a)  Find  the  sum  of  the  nine  decimals. 

*  Every  common  fraction  can  be  changed  to  hundredths  by  annexing  two  zeros 
to  the  numerator  and  dividing  by  the  denominator;  but  this  method  of  reduction 
is  not  always  the  most  simple. 


256  COMPLETE    ARITHMETIC. 

Fractions. 
205.  Denominate  Fractions. 

1.  One  half  inch  is  what  part  of  a  foot  ? 

2.  Two  and  ^  inches  are  what  part  of  a  foot  ? 

3.  Five  and  ^  inches  are  what  part  of  a  foot  ? 

4.  One  half  foot  is  what  part  of  a  rod  ? 

5.  Three  and  one  half  feet  are  what  part  of  a  rod  ? 

6.  Ten  and  one  half  feet  are  what  part  of  a  rod  ? 

7.  Sixty-four  rods  are  what  part  of  a  mile  ? 

8.  Mnety-six  rods  are  what  part  of  a  mile  ? 

9.  One  hundred  eighty  rods  are  what  part  of  a  mile  ? 

10.  One  and  one  half  quarts  are  what  part  of  a  peck  ? 

11.  Two  and  one  half  quarts  are  what  part  of  a  gallon  ? 

12.  Twenty-four  quarts  are  what  part  of  a  bushel  ? 

13.  Fourteen  ounces  are  what  part  of  a  pound  ? 

14.  Seven  and  one  half  ounces  are  what  part- of  a  pound? 

15.  One  and  one  fourth  ounces  are  what  part  of  a  pound? 

16.  Six  hundred  pounds  are  what  part  of  a  ton  ? 

17.  Four  hundred  fifty  pounds  are  what  part  of  a  ton  ? 

18.  Six  hundred  twenty-five  pounds  are  what  part  of  a  ton  f 

19.  Seventy-five  square  rods  are  what  part  of  an  acre  ? 

20.  Forty -five  square  rods  are  what  part  of  an  acre  ? 

21.  One  hundred  square  rods  are  what  part  of  an  acre  ? 

22.  Thirty-two  cubic  feet  are  what  part  of  a  cord  ? 

23.  Fifty-six  cubic  feet  are  what  part  of  a  cord  ? 

24.  One  hundred  cubic  feet  are  what  part  of  a  cord  ? 

25.  Seven  and  one  half  minutes  are  what  part  of  an  hour  ? 

26.  Forty  minutes  are  what  part  of  an  8-hour  day  ? 

27.  Ninety  minutes  are  what  part  of  an  8-hour  day  ? 


PART   II.  257 

Algebraic  Fractions. 
206.  Fractions  in  Equations. 
Example  I. 

2 

Multiplying  both  members  of  the  equation  (see  page  217,  Art. 
170,  Statement  4)  by  2,  the  denominator  of  the  fraction  in  the  equa- 

a?  +  4a?  =  60 
Uniting  terms,  5a;  =  60 

Dividing  by  5,  jc  =  12 

Example  II. 

f +  f +  5a;  =  70 
2      3 

2r 
Multiplying  by  2,        x -^  —  -f  IQx  =  140 

o 
Multiplying  by  3,      Zx-^2x-\-  30aj  =  420* 
Uniting  terms,  35ic  =  420 

Dividing  by  35,  ic  =  12 


Problems. 

Find  the  value  of  x. 

1.  ^+^ 

5       3 

=  8. 

3.  -  +  5a;  =  64. 
3 

^      X         X 

=  6. 

4.  2a;-- =  50. 

*  Observe  that  the  equation  might  have  been  cleared  of  fractions  by  multiplying 
both  its  members  by  6,  the  1.  c.  m.  of  2  and  3. 


258  COMPLETE    ARITHMETIC. 

Algebra. 

207.  Problems   Leading   to   Equations   Containing  One 
Unknown  Quantity,  Without  Fractions. 

Example. 
John  and  Henry  together  have  60  oranges,  and  Henry 
has  three  times  as  many  as  John.     How  many  has  each  ? 
Let  X  =  the  number  John  has, 

then        Sx  =  the  number  Henry  has, 
and  x-\-Zx  —  the  number  they  together  have. 
Therefore  x-\-Sx=  60. 
Uniting  4:X—  60. 

Dividing  x  =  16. 

Multiplying     3x  =  45. 
John  has  15  oranges  and  Henry  has  45  oranges. 

Problems. 

1.  The  sum  of  two  numbers  is  275,  and  the  greater  is 
four  times  the  less.     What  are  the  numbers  ? 

2.  Eobert  has  a  certain  sum  of  money  and  Harry  has  five 
times  as  much;  together  they  have  $216.  How  many  dol- 
lars has  each  ? 

3.  One  number  is  four  times  another,  and  their  difference 
is  270.     What  are  the  numbers  ? 

4.  Peter  has  a  certain  number  of  marbles  and  William 
has  8  more  than  Peter;  together  they  have  96  marbles. 
How  many  has  each  ? 

5.  Sarah  has  a  certain  number  of  pennies  and  her  sister 
has  nine  more  than  twice  as  many;  together  they  have  93. 
How  many  has  each  ? 

6.  Two  times  Keuben's  money  plus  three  times  his  money 
equals  175  dollars.     How  many  dollars  has  he  ? 


PART    II.  259 

Qeometry. 
208.  Construction  Problems — Triangles. 


1.  Draw  a  triangle  whose  base,  ah,  is  3  inches  long.  Make 
the  angle  a,  55°  and  the  angle  h,  35°.  The  angle  c  should 
be  degrees.     Measure  the  three  sides. 

2.  Draw  a  triangle  two  of  whose  sides  are  equal.  Measure 
and  compare  the  angles  opposite  the  equal  sides. 

Observe  that  a  triangle,  two  of  whose  sides  are  equal,  has 
two  angles  equal;  and  conversely  if  two  angles  of  a  triangle 
are  equal,  two  of  the  sides  are  equal. 

3.  If  two  triangles  have  the  three  sides  of  one  equal  to  the 
three  sides  of  the  other,  each  to  each,  do  you  think  the  two 
triangles  are  alike  in  every  respect  ? 

4.  If  two  triangles  have  the  three  angles  of  one  equal  to 
the  three  angles  of  the  other,  each  to  each,  do  you  think  the 
two  triangles  are  necessarily  alike  iji  every  respect  ? 

5.  Draw  two  triangles,  the  angles  of  one  being  equal  to 
the  angles  of  the  other,  and  the  sides  of  one  not  being  equal 
to  the  sides  of  the  other. 

6.  Is  it  possible  to  draw  a  triangle  whose  sides  are  equal, 
but  whose  angles  are  unequal  ? 

7.  Is  it  possible  to  draw  a  quadrilateral  whose  sides  are 
equal  but  whose  angles  are  unequal  ? 


260  COMPLETE    ARITHMETIC. 

209.  Miscellaneous  Review. 

1.  Without  a  pencil,  change  each  of  the  following  frac- 
tions to  hundredths :     -,    1,   A,    1,    1,   1,    A,   _I_,   A, 

5       8       8      10     25     50     12i     33^     16| 
60       30       45       60       60       60       60 
500'    200'    900'    150'    250'    125'    66f* 

2.  Butter  that  cost  25^  a  pound  was  sold  for  29^  a  pound. 
The  gain  was  equal  to  what  part  of  the  cost  ?  The  gain  was 
equal  to  how  many  hundredths  of  the  cost  ? 

3.  The  taxes  on  an  acre  of  land  which  was  valued  at  $600 
were  $12.  The  taxes  were  equal  to  what  part  of  the 
valuation  ?  The  taxes  were  equal  to  how  many  hundredths 
of  the  valuation  ? 

4.  Mr.  Jones  purchased  500  barrels  of  apples.  He  lost 
by  decay  a  quantity  equal  to  75  barrels.  What  part  of  his 
apples  did  he  lose  ?  How  many  hundredths  of  his  apples 
did  he  lose  ? 

5.  Kegarding  a  month  as  30  days  and  a  year  as  360  days, 
what  part  of  a  year  is  7  months  and  10  days  ?  How  many 
hundredths  of  a  year  in  7  months  and  1 0  days  ?  How  many 
thousandths  of  a  year  ?  How  many  ten-thousandths  of  a 
year? 

6.  One  cord  48  cubic  feet  is  what  part  of  4  cords  16  cubic 
feet  ?  Change  the  fraction  to  hundredths  ;  to  thousandths ; 
to  ten-thousandths. 

7.  One  mile  240  rods  is  what  part  of  3  miles  160  rods? 
Change  the  fraction  to  hundredths ;  to  thousandths ;  to  ten- 
thousandths. 

8.  From  a  bill  of  $175  there  was  a  discount  of  $14.  The 
discount  is  equal  to  how  many  hundredths  of  the'  amount  of 
the  bill? 


PEECENTAGE. 

210.  Per  cent  means  hundredth  or  hundredths.  Per  cent 
may  be  expressed  as  a  common  fraction  whose  denominator 
is  100,  or  it  may  be  expressed  decimally ;  thus,  6  per  cent  = 

j-f-g-  or  .06;  28i-  per  cent  =  j^^  or  .28|-;  |-  per  cent  =  j^g 

or  .00|. 

Note. — Instead  of  the  words  per  cent  sometimes  the  sign  (%)  is 
used ;  thus  6  per  cent  may  be  written  6  %. 

211.  The  base  in  percentage  is  the  number  of  which 
hundredths  are  taken ;  thus,  in  the  problem,  find  1 1  %  of 
600,  the  base  is  600 ;  in  the  problem,  16  ^s  what  per  cent  of 
800  ?  the  base  is  800  ;  in  the  problem,  18  is  3%  of  what? 
the  base  is  not  given,  but  is  to  be  found  by  the  student. 

Observe  that  whenever  the  base  is  given  in  problems  like  the  above,  it 
follows  the  word  of. 

212.  There  are  three  cases  in  percentage  and  only  three. 

Case  I.  To  find  some  per  cent  (hundredths)  of  a  number, 
as:  find  15%  of  600. 

Case  II.  To  find  a  number  when  some  per  cent  of  it  is 
given,  as  :  24  is  8%  of  what  number  ? 

Case  III.  To  find  what  per  cent  one  number  is  of  another, 
as :  12  is  what  %  of  400  ? 

Observe  that  a  thorough  knowledge  of  fractions  is  the  necessary 
preparation  for  percentage.  The  work  in  percentage  is  work  in 
fractions,  the  denominator  employed  being  100. 

261 


262  COMPLETE    ARITHMETIC. 

Percentage. 

213.  Case  I. 

Find  17  per  cent  (.17)  of  8460. 

Note. — We  may  find  f  of  a  number  by  finding  3  times  1  fourth 

of  it;    that  is,  by  multiplying  it  by  |.     So  we  may  find  .17  of  a 

number  by  finding  17  times  1  hundredth  of  the  number;  that  is,  by 

multiplying  by  .17. 

Operation.  Explanation. 

84^60 

.17  One  per  cent  (1  hundredth)  of  8460  is  84.60;  17 

592  20      P®^  ^^"*  (hundredths)  of  8460  is  17  times  84.60,  br 
846^0        1438.20. 


1438.20 


Problems. 

1.  Find  17%  of  6420 ;  of  5252  ;  of  31.40. 

2.  Find  35%  of  6420 ;  of  5252  ;  of  31.40. 

3.  Find  43%  of  6420;  of  5252  ;  of  31.40. 

4.  Find  25%  of  6420;  of  5252;  of  31.40. 

5.  Find  50%  of  6420 ;  of  5252  ;  of  31.40. 

6.  Find  30%  of  6420;  of  5252  ;  of  31.40. 

7.  Find  35%  of  6420;  of  5252  ;  of  31.40. 

8.  Find  65%  of  6420;  of  5252  ;  of  31.40. 
(a)  Find  the  sum  of  the  twenty-four  results. 

9.  A  sold  goods  for  B.  As  remuneration  for  his  services 
he  received  a  sum  equal  to  12  %  of  the  sales.  He  sold  $2146 
worth  of  goods.     How  much  did  he  receive  ? 

10.  C  is  a  collector  of  money.  For  this  service  he  charges 
a  commission  of  6  %  ;  that  is,  his  pay  is  6  %  of  the  amount 
collected.  He  collected  for  D  $375.  How  much  should  he 
pay  over  to  D,  and  how  much  should  he  retain  as  pay  for 
collecting  ? 


PART    II. 


263 


Percentage. 

214.  Case  II. 

673.20  is  17  per  cent  (.17)  of  what  number? 


Operation  iN'o.  1. 
17)673.20(39.60 
51  100 


163 
153 


3960. 


102 
102 


Explanation, 

Since  673.20  is  17  hundredths  of  the 
number,  1  hundredth  of  the  number  is 
1  seventeenth  of  673.20,  or  39.60 ;  and 
100  hundredths  =  100  times  39.60,  or 
3960. 


Operation  No.  2. 
.17)673.20X3960 
51 


163 
153 

102 
102 


Explanation. 

We  may  find  100  seventeenths  of  a 
number  by  finding  1  seventeenth  of  a 
hundred  times  the  number.  100  times 
673.20  is  67320.  1  seventeenth  of  67320 
is  3960. 


Observe  that  dividing  by  .17  is  finding  ^^  of  the  dividend,  just 
as  dividing  by  |  is  finding  |  of  the  dividend,  and  dividing  by  ^  is 
finding  f  of  the  dividend. 

Note. — Sometimes  the  process  may  be  shortened  by  writing  the 
per  cent  as  a  common  fraction  and  reducing  it  to  its  lowest  terms ; 
then  using  the  reduced  fraction  instead  of  the  one  whose  denomina- 
tor is  100. 

Problems. 

1.  360  is  15%  of  what  number? 

2.  360  is  25%  of  what  number? 

3.  360  is  50%  of  what  number? 

4.  360  is  75%  of  what  number? 

5.  360  is  40%  of  what  number? 
(a)  Find  the  sum  of  the  five  resuliis. 


264  COMPLETE    ARITHMETIC. 

Percentage. 

Case  II. — Continued. 
Problems. 

1.  $34.32  is  13  per  cent  of  what? 

2.  $34.32  is  15  per  cent  of  what? 

3.  $34.32  is  25  per  cent  of  what? 

4.  $34.32  is  33i  per  cent  of  what  ? 

5.  $34.32  is  50  per  cent  of  what  ? 

6.  $64.98  is  19  per  cent  of  what? 

7.  $64.98  is  12  per  cent  of  what  ? 

8.  $64.98  is  20  per  cent  of  what  ? 

9.  $64.98  is  24  per  cent  of  what  ? 

10.  $64.98  is  25  per  cent  of  what  ? 
(a)  Find  the  sum  of  the  ten  results. 

11.  A  sold  goods  for  B.  As  remuneration  for  his  services 
A  received  a  sum  equal  to  12%  of  the  sales.  He  received 
$33.06.  What  was  the  amount  of  his  sales  ?  How  much 
money  does  B  have  left  of  what  he  received  for  the  goods 
after  paying  out  of  it  A's  commission  ? 

12.  C  is  a  collector  of  money.  For  this  service  he  charges 
a  commission  of  6  %  ;  that  is,  his  pay  is  6  %  of  the  amount 
collected.  His  commission  on  a  certain  collection  was 
$74.40.  What  was  the  amount  collected  ?  How  much 
should  the  man  for  whom  he  collected  the  money,  receive  ? 

13.  C  collected  a  sum  of  money  for  D,  deducted  his  com- 
mission of  6%,  and  paid  the  remainder  of  the  sum  collected 
to  D.  He  paid  D  $350.15.*  What  was  the  sum  collected  ? 
How  much  money  did  C  retain  as  his  commission  for  col- 
lecting ? 

* 8350.15  is  what  %  of  the  amount  collected? 


PART   II. 


26b 


Operation  No.  1. 

)625"^( 
5400 


900)625"^00(.69| 


8500 
8100 


400     4 


Percentage. 

215.  Case  III. 

625  is  what  per  cent  of  900  ? 

Explanation, 

625  is  lit  of  900. 

This  fraction  may  be  changed  to 
hundredths  in  the  usual  manner,  viz., 
by  performing  the  division  indicated^ 
"  carrying  out "  to  hundredths  only. 


900     9 


Operation  No. 

2. 

Explanation. 

625   _   25 

36)25^00(.69-|  = 

:69|< 

% 

625  is  f  §§  or  H  of  900. 

216 

The    fraction,    ||,    may  be 

340 

changed  to  hundredths  in  the 

324 

usual  manner. 

16      4 

36  ~  9 

Operation 

and  Explanation  No.  3. 

625 

625  is  -—  of  900. 

900 

625- 
900- 

\:-z- "•-">' 

Operation  and  Explanation  No.  4. 
One  hundredth  of  900  is  9 ;  then  625  is  as  many  hundredths  of 
900  as  9  is  contained  times  in  625.    9  is  contained  in  625,  69f  times ; 
so  625  is  Q9i-%  (hundredths)  of  900. 

Problems. 
What  per  cent  (how  many  hundredths)  of  800  is — 
(1)250?       (2)375?       (3)475?       (4)350?      (5)150? 
(a)  Find  the  sum  of  the  five  results. 


266  COMPLETE    ARITHMETIC. 


Percentage. 

Case  III. — Continued. 

Problems. 

What  %  is— 

1.     32  of  600  ?          6.     50  of  750  ? 

11. 

40  of  325  ? 

2.     54  of  600?          7.     90  of  750? 

12. 

50  of  325  ? 

3.     75  of  600  ?          8.     85  of  750  ? 

13. 

62  of  325  ? 

4.     95  of  600  ?          9.     80  of  750  ? 

14. 

78  of  325  ? 

5.  344  of  600?        10.  445  of  750? 

15. 

95  of  325? 

(a)  Find  the  sum  of  the  first  five  results. 

(b)  Find  the  sum  of  the  second  five  results. 

(c)  Find  the  sum  of  the  third  five  results. 

What  %  is — 

1 6.  $.76  of  $38  ?  (What  is  one  %  of  $38 ?) 

17.  $9.02  of  $225.50  ?  (What  is  one  %  of  $225.50?) 

18.  $17.13  of  $342.60  ?  (What  is  one  ^,of  $343.60?) 

19.  $58.45  of  $835  ?  (What  is  one  %  of  $835?) 

20.  $1.29  of  $6.45  ?  (What  is  one  %  of  $6.45?) 

21.  $113.10  of  $754  ?  (What  is  one  %  of  $754?) 

22.  $21  of  $175  ?  (What  is  one  %  of  $175?) 

23.  A  sold  goods  for  B  to  the  amount  of  $346.25 ;  he 
received  as  his  commission  for  selling  the  goods,  $41.55. 
His  commission  was  what  per  cent  of  the  sales  ? 

24.  C  collected  for  D  $643.50 ;  he  retained  as  his  com- 
mission for  collecting  the  money,  $38.61.  His  commission 
was  what  per  cent  of  the  sales  ? 

25.  F  purchased  256  barrels  of  apples;  he  lost  by  decay 
a  quantity  equal  to  16  barrels.  What  %  of  his  apples  did 
he  lose  ? 


PART  II.  267 

Algebra. 

216.  Problems  Leading  to  Equations  Containing 
Fractions. 

If  to  i  of  Bemie's  money  you  add  ^  of  his  money  the 
sum  is  $63.     How  much  money  has  he  ? 

Let  X  =  the  number  of  dollars  he  has. 


Then  |  + 

X 

=  63. 

Multiplying  by  3, 

Zx 

-+4 

=  189 

Multiplying  by  4,         4:  x  -\-  S  x  =  756* 
Uniting  terms,                          7  £C  =  756 
Dividing  by  7,                              (K  =  108 

Thus  we  find  that  Bemie  had  108  dollars. 

1.  If  to  ^  of  a  certain  number  you  add  \  of  the  same 
number,  the  sum  is  45.     What  is  the  number  ? 

2.  A  lady  upon  being  asked  her  age  replied :  If  from  ^  of 
my  age  you  subtract  \  of  my  age  the  remainder  will  be  9 
years.     How  old  was  she  ? 

3.  The  sum  of  two  numbers  is  72,  and  the  less  number 
equals  |-  of  the  greater  number.     What  are  the  numbers  ? 

Let  X  =  the  greater  number. 

4.  The  sum  of  two  numbers  is  75,  and  the  less  number 
equals  f  of  the  greater  number.     What  are  the  numbers  ? 

5.  The  difference  of  two  numbers  is  30,  and  the  less 
number  equals  |  of  the  greater  number.  What  are  the 
numbers  ? 

*  Observe  that  the  equation  might  have  been  cleared  of  fractions  by  multiplying 
both  its  members  by  12,  the  1.  c.  m.  of  3  and  4. 


268  COMPLETE   ARITHMETIC. 

Algebra. 

217.  Miscellaneous  Problems. 
1.  Harry  has  some  marbles;  Joseph  has  8  more  than  i  as 
many  as  Harry;  together  they  have  68.     How  many  has 
each  ? 

.  2.  William  has"  12  more  than  i  as  many  cents  as  Lucius; 
together  they  have  87  cents.     How  many  cents  has  each  ? 

3.  If  to  the  half  of  a  certain  number  you  add  a  third  of 
the  number,  the  sum  will  be  5  more  than  3  fourths  of  the 
number.     What  is  the  number  ? 

Let  X  =  the  number, 

Then  -A 5  =  — 

2      3  4 

4.  If  to  1^  of  a  number  you  add  -i^  of  the  same  number, 
the  sum  will  be  12  more  than  |  of  the  number.  What  is 
the  number  ? 

5.  If  to  a  certain  number  you  add  5  times  the  number, 
the  sum  will  be  24  more  than  4  times  the  number.  What 
is  the  number  ? 

6.  If  to  |-  of  a  number  you  add  i  of  the  number  the  sum 
will  be  a.     What  is  the  number  ? 

Let  X  =  the  number. 

Then  -  +  -  =  a 
2       3 

Multiplying  by  6,         3«  +  2a3  =  6a 
Uniting  6x  =  6a 

Dividing  by  5  x  =  — 

5 

Observe  that  yoii  may  put  in  the  place  of  a  any  number  you  please ; 
hence  any  number  is  g  of  the  sum  of  its  half  and  third. 


PART    II.  269 

Geometry. 
218.  Construction  Problems — Quadrilaterals. 


1.  Draw  a  rhombus.  Make  the  angle  a,  110°.  Use  the 
protractor  for  measuring  the  angle  a  only.  Make  ah  and 
ac  each  2  inches.  Then  draw  hd  parallel  to  ac  and  cd  paral- 
lel to  ab.     Prove  your  work  by  measuring  the  other  angles. 

Ohserve  that  when  one  side  and  one  angle  of  a  rhombus 
are  given  the  rhombus  may  be  drawn. 

2.  If  the  angle  a  of  a  rhomboid  is  110°,  how  many  de- 
grees in  angle  b  ?     In  angle  c  ?     In  angle  d  ? 

'  3.  Draw  a  trapezoid.  Make  the  side  a6  3  inches  long. 
Make  the  angle  a,  110  degrees,  and  the  line  ac  2  inches  long. 
Make  the  angle  &,  90  degrees ;  and  the  line  bd  of  indefinite 
length,  but  long  enough  to  form  the  side  bd.  Draw  cd  par- 
allel to  ab.  Prove  the  work  by  finding  the  sum  of  the  four 
angles. 

4.  Draw  a  trapezium.  Make  the  angle  a,  100° ;  the  line 
ab  3  inches  long,  and  the  line  ac  2  inches  long.  Make  the 
angle  b,  120°,  and  the  side  bd  of  indefinite  length.  Make 
the  angle  c,  85°.  Angle  d  should  be  an  angle  of  how  many 
degrees  ?     Prove  the  work  by  measuring  angle  d. 

5.  Draw  four  lines  that  together  make  a  very  irregular 
trapezium ;  then  using  the  protractor,  find  the  sum  of  the 
four  angles. 


270  COMPLETE    ARITHMETIC. 

219.  Miscellaneous  Review. 

1.  If  I  lose  ^\  of  my  money/what  %  of  my  money  do  I 
lose? 

2.  Mr.  Button  spent  $5  of  the  $12  which  he  had  earned. 
What  per  cent  of  what  he  earned  did  he  spend  ? 

3.  Mr.  Thomas  earns  $150  per  month.  The  monthly 
rent  of  the  house  in  which  he  lives  is  equal  to  15%  of  what 
he  earns.     How  much  is  his  rent  per  year  ? 

4.  Mr.  Jones  purchased  500  barrels  of  apples.  He  lost 
by  decay  a  quantity  equal  to  75  barrels.  What  per  cent  of 
the  apples  purchased  remained  sound  ? 

5.  Mr.  Brown  borrowed  $625.  He  used  87%  of  this 
money  to  pay  debts.  How  much  of  the  money  borrowed 
did  he  have  left  ? 

6.  Mr.  Green  paid  $24.60  for  a  suit  of  clothes.  If  this 
was  exactly  15%  of  his  monthly  earnings,  how  much  does 
he  earn  per  month  ? 

7.  Mr.  White  earns  $1500  per  year.  He  spends  $312  for 
board,  $200  for  clothes,  $75  for  books  and  papers,  $80  for 
traveling  and  amusements;  gives  to  his  church  $40,  for 
charitable  purposes,  $35,  and  all  his  other  expenses  amount 
to  $23.  The  remainder  of  his  salary  he  puts  into  a  savings 
bank.     What  per  cent  of  his  salary  does  he  save  ? 

8.  If  to  ^  of  my  money  you  add  ^  of  my  money  the  sum 
will  be  $63.     How  much  money  have  I  ? 

9.  The  sum  of  two  numbers  is  120,  and  the  less  number 
is  "I  of  the  greater.     What  are  the  numbers  ? 

10.  If  one  angle  of  a  rhomboid  is  80°,  what  is  the  size  of 
each  of  the  other  angles  ? 

11.  Three  twenty-fifths  is  what  %  of  f  ? 


PEECENTAGE. 

220.   Problems  in  Case  III,  somewhat  disguised. 

Example  No.  1. 

75  is  how  many  %  more  than  60  ? 

75  is  15  more  than  60 ;  so  the  percentage  part  of  this  problem  is, 
15  is  what  %  of  601  Ans.  25^.  Therefore  75  is  25^  more  than  60  i 
that  is  75  is  25  hundredths  of  60  more  than  60. 

Example  No.  2. 
60  is  how  many  %  less  than  75  ? 

60  is  15  less  than  75  ;  so  the  percentage  part  of  this  problem  is,  15 
is  what  %  of  75?  Ans.  20^.  Therefore  60  is  20^  less  tharr  75; 
that  is,  60  is  20  hundredths  of  75  less  than  75. 

Observe  that  in  example  No.  1, 60  is  the  base,  and  that  in  example 
No.  2,  75  is  the  base ;  and  that  in  problems  of  this  kind  the  base 
always  follows  the  word  than. 

Problems. 

1.  60  is  how  many  per  cent  more  than  45  ? 

2.  45  is  how  many  per  cent  less  than  60  ? 

3.  150  is  how  many  per  cent  more  than  125  ? 

4.  125  is  how  many  per  cent  less  than  150  ? 

5.  225  is  how  many  per  cent  more  than  200  ? 

6.  200  is  how  many  per  cent  less  than  225  ? 

7.  James  has  $345 ;  Peter  has  $414.  Peter  has  how 
many  per  cent  more  than  James?  James  has  how  many 
per  cent  less  than  Peter  ? 

271 


272  COMPLETE   ARITHMETIC. 

Percentage. 
221.  Application  of  Art.  220  to  "Loss  and  Gain." 

Find  the  per  cent  of  loss  or  gain  in  each  of  the  following:* 

1.  Bought  for  25^  and  sold  for  30^. 

2.  Bought  for  25^  and  sold  for  23^. 

3.  Bought  for  25^  and  sold  for  27^. 

4.  Bought  for  25^  and  sold  for  21^. 

5.  Bought  for  $40  and  sold  for  $48. 

6.  Bought  for  $40  and  sold  for  $35. 

7.  Bought  for  $40  and  sold  for  $52. 

8.  Bought  for  $40  and  sold  for  $36. 

9.  Sold  for  65^  that  which  cost  50^. 
10.  Sold  for  18^  that  which  cost  20^. 
IJ.  Sold  for  30^  that  which  cost  36f 

12.  Sold  for  35^  that  which  cost  30^. 

13.  Sold  for  $50  that  which  cost  $40. 

14.  Sold  for  $40  that  which  cost  $50. 

15.  Sold  for  $30  that  which  cost  $40. 

16.  Mr.  Watson  bought  60  lbs.  of  tea  at  32^  a  pound  and 
sold  it  at  50^.  What  was  his  per  cent  of  gain  if  he  sold  as 
many  pounds  as  he  bought  ?  If  he  lost  4  lb.  by  "  down- 
weights  "  and  wastage,  how  much  money  did  he  gain  ? 
What  was  his  real  per  cent  of  gain  ? 

17.  Mr.  Jenkins  bought  gloves  at  $4.50  per  dozen  and 
sold  them  at  50^  a  pair.     What  was  his  per  cent  of  gain  ? 

18.  Mr.  Warner  bought  apples  at  40^  a  bushel.  He  lost 
25%  of  them  by  decay  and  sold  the  remainder  at  50^  a 
bushel.  Did  he  gain  or  lose  by  the  transaction  ?  What  was 
his  per  cent  of  gain  or  loss  ? 

*  In  speaking  of  the  per  cent  of  loss  or  of  gain  the  cost  is  regarded  as  the  hose 
unless  otherwise  specified. 


PART   II.  273 

Percentage. 
222.  Percentage  Problems  under  Case  I,  in  which  the 

PER    cent   is   more    THAN    100. 

Find  175%  (III  or  1.75)  of  $632.60. 
Operation  No.  1.  Explanation. 

$6'32.60  One  %  of  ^632.60  is  16.326. 

1.75  175^   of   1632.60    is    175    times  |6.326.  or 

$31.6300      '^1107.05. 
$442  820  Observe  that  b%  of  $632.60  is  131.63;  that 

$632  60  "^^^  °^  $632.60  is  1442.82 ;  that  100^  of  $632.60 

is  $632.60. 


$1107.0500 


Operation  No.  2. 
175%  =  \U  =  T-     i  of  $632.60  =  $158.15. 
I  of  $632.60  =  7  times  $158.15,  or  $1107.05. 

Problems. 

1.  Find  175%  of  356;  of  276;  of  540.20. 

2.  Find  155%  of  356;  of  276;  of  540.20. 

3.  Find  145%  of  356;  of  276;  of  540.20. 

4.  Find  125%  of  356;  of  276;  of  540.20. 

5.  Find  200%  of  356;  of  276;  of  540.20. 

6.  Find  150%  of  356;  of  276;  of  540.2a. 

7.  Find  250%  of  356 ;  of  276 ;  of  540.20. 
(a)  Find  the  sum  of  the  twenty-one  results. 

8.  David's  money  is  equal  to  150%  of  Henry's  money. 
Henry  has  $240.     How  much  has  David  ? 

9.  If  goods  cost  $260,  and  the  profit  on  them  is  125%, 
what  is  the  selHng  price  ? 

10.  If  a  Chicago  lot  at  the  beginning  of  a  certain  year  was 
worth  $8000,  and  during  the  year  increased  in  value  250%, 
what  was  it  worth  at  the  end  of  the  year  ? 


274 


COMPLETE    ARITHMETIC. 


Percentage. 
223.    Percentage   Problems   under   Case   II,  in  which 

THE   PER   CENT   IS   MORE   THAN    100. 

$1107.05  is  175%  of  how  much  money? 


Operation  No.  1. 
175mi07\05(S6.326 


1050 

~570 
525 


100 


$632.60 


455 
350 

1050 
1050 


Explanation, 


Since  $1107.05  is  175  hundredths 
of  the  money,  1  hundredth  of  the 
money  is  one  176th  of  $1107.05,  or 
$6,326;  and  XOO  hundredths,  100 
times  16.326,  or  $632.60. 


Operation  No.  2. 
1.75W107.05X$632.60 
1050 

570 
525 


455 
350 


1050 
1050 


Explanation. 
We  may  find  one  hundred  175ths 
of  a  number  by  finding  one  175th  of 
100  times  the  number.  One  hundred 
times  $1107.05  is  $110705.  One  175th 
of  $110705  is  $632.60. 

Operation  and  Explanation  No.  3. 
175^  =  ig§  =  |.    If  $1107.05  is  7  fourths  of 
the  money,  1  seventh  of  $1107.05,  or  $158.15,  is 
1  fourth  of  the  money,  and  4  fourths,  or  the 
whole  of  it,  is  4  times  $158.15,  or  $632.60 

Problems. 


1.  43.50  is  125%  of  what  number? 

2.  65.10  is  150%  of  what  number? 

3.  48.50  is  200%  of  what  number? 

4.  59.20  is  250%  of  what  number? 

5.  29.44  is  115%  of  what  number? 

(a)  Find  the  sum  of  the  five  answers. 


PART    II.  275 

Percentage. 

224.    Percentage   Problems   under   Case  III,  in  which 

THE  PER  CENT  IS  MORE  THAN  100. 

$1107.05  is  what  per  cent  of  $632.60  ? 

Operation.  Explanation. 

$6.326)$1107.050X175 

6326  One  per  cent  of  $632.60  is  $6,326  ; 

AHAA.^  then  $1107.05  is  as  many  per  cent  of 


44282 


$632.60  as  $6,326  is  contained  times 
in  $1107.05.  It  is  contained  175  times, 

^1630  so  ^1107.05  is  175^  of  632.60. 

31630 

Note. — The  above  problem  may  be  solved  as  a  similar  problem  is 
solved  on  page  265,  operation  No.  1.  $1107.05  is  HB:U  of  $632.60. 
Perform  the  division  indicated,  carrying  out  to  hundredths  only.  The 
quotient  will  be  1.75  or  i§^  =  175^. 

Problems. 

1.  What  per  cent  of  845  is  2112.5  ? 

2.  What  per  cent  of  845  is  1056.25  ? 

3.  What  per  cent  of  845  is  1352  ? 

4.  What  per  cent  of  845  is  1183  ? 

5.  What  per  cent  of  845  is  2746.25  ? 
(a)  Find  the  sum  of  the  5  answers. 

6.  Eeuben  has  $2420  ;  Bernie  has  $4961.  Bernic's  money 
equals  how  many  per  cent  of  Eeuben's  money  ? 

7.  A  certain  house  cost  $6425.  The  lot  upon  which  it 
stands  cost  $2325.  (a)  The  cost  of  the  house  equals  how 
many  per  cent  of  the  cost  of  the  lot  ?  (b)  The  cost  of  the 
lot  equals  how  many  per  cent  of  the  cost  of  the  house  ? 


276  COMPLETE    ARITHMETIC. 

Percentage. 
225.  Miscellaneous  Problems. 

1.  Find  1-%*  of  632  ;  of  356 ;  of  272. 

2.  Find  |%  of  632  ;  of  356  ;  of  272. 

3.  Find  .25 %t  of  632  ;  of  356  ;  of  272. 

(a)  Find  the  sum  of  the  nine  results. 

4.  Find  3|-%  of  496 ;  of  532  ;  of  720. 

5.  Find  4J%  of  496 ;  of  532  ;  of  720. 

6.  Find  1.75%  of  496  ;  of  532  ;  of  720. 

(b)  Find  the  sum  of  the  nine  results. 

7.  What  part  of  94  is  11  ?  J     What  per  cent  ? 

8.  What  part  of  94  is  36  ?     What  per  cent  ? 

9.  What  part  of  94  is  47  ?     What  per  cent  ? 

(c)  Find  the  sum  of  the  three  "  per  cents." 

10.  15  is  3%  of  what  number  ?     4%  ?     5%  ? 

11.  24  is  3%  of  what  number  ?     4%  ?     5%  ? 

12.  42  is  3%  of  what  number  ?     4%  ?     5%  ? 

(d)  Find  the  sum  of  the  nine  answers. 

13.  375  is  125%  (||-|-)  of  what  number? 

14.  436  is  109%  (Iff)  of  what  number? 

15.  598  is  115%  (|i|)  of  what  number? 

(e)  Find  the  sum  of  the  three  answers. 

16.  544  is  15%  less  than  what  number? 

17.  545  is  25%  more  than  what  number? 

18.  510  is  170%  of  what  number  ? 

(f)  Find  the  sum  of  the  three  answers. 

*  This  means  J  of  1  per  cent. 

t  This  means  find  25  hundredths  of  1  per  cent. 

t  Answer  with  a  common  fraction  in  its  lowest  terms. 


PART    II.  277 

Algebra. 

226.   To  Find  Two  Numbers  when  their  Sum  and 
Difference  are  Given. 

1.  The  difference  of  two  numbers  is  4  and  their  sum  is  20. 
What  are  the  numbers  ? 

Let  X  —  the  smaller  number, 

then         x-\-  Ai  —  the  larger  number, 
and  a?  +  ic  +  4  -  20. 

Transposing,  ^  +  ^  =  20  —  4. 

Uniting,  2  a?  =  16. 

Dividing,  a?  =  8,  the  smaller  number. 

ic  +  4  —  12,  the  larger  number. 

2.  The  difference  of  two  numbers  is  9  and  their  sum  is 
119.     What  are  the  numbers  ? 

3.  The  difference  of  two  fractions  is  ^V  ^^^  their  sum  is 
\.     What  are  the  fractions  ? 

4.  The  difference  of  two  numbers  is  d  and  their  sum  is  s. 
What  are  the  numbers  ? 

Let  X  =  the  smaller  number, 

then         X  -\-  d  =  the  larger  number, 
and  X  -\-  X  -\-  d  =  s 

Transposing,  x -\-  x  =  s  —  d 

Uniting,  2x  =  s  —  d 

Dividing,  x  = 


2 

Observe  that  any  number  you  please  may  be  put  in  the  place  of  s, 
and  any  number  less  than  s  in  the  place  of  d  ;  so  when  the  sum  and 
the  difference  of  two  numbers  are  given,  the  smaller  number  may  be 
found  by  subtracting  the  difference  from  the  sum  and  dividing  the 
remainder  by  2. 


278  COMPLETE    ARITHMETIC. 

Algebra. 

227.  Another  Method  of  Finding  Two  Numbers  when 
THEIR  Sum  and  Difference  are  Given. 

1.  The  difference  of  two  numbers  is  17  and  their  sum  is 
69.     What  are  the  numbers  ? 

Let  X  =  the  larger  number, 

then         X  —  11  =  the  smaller  number, 
and  x-\-  X  —  11  =  ^^. 

Transposing,     x -\-  x  =  69  +  17. 

Uniting,  2x  =  86. 

Dividing,  ic  —  43,  the  larger  number. 

X  —  11  =  2^,  the  smaller  number. 

2.  The  difference  of  two  numbers  is  8.4  and  their  sum 
75.6.     What  are  the  numbers  ? 

3.  The  difference  of  two  numbers  is  d  and  their  sum  is  s. 
What  are  the  numbers  ? 

Let  a?  r=  the  larger  number, 

then  X  -  d  —  the  smaller  number, 

and     X  -\-  X  —  d  =  s 

Transposing,      x  -\-  x  =  s  -\-  d 

Uniting,  2^  =  s  +  c? 

s^d 
Dividing,         •  x  — 

^'  2 

Observe  that  any  number  you  please  may  be  put  in  the  place  of  s, 
and  any  number  less  than  s  in  the  place  of  d ;  so  when  the  sum  and 
difference  of  two  numbers  are  given,  the  larger  may  be  found  by  add- 
ing the  difference  to  the  sum  and  dividing  the  amount  by  2. 

4.  A  horse  and  a  harness  together  are  worth  $146,  and 
the  horse  is  worth  $74  more  than  the  harness.  Find  the 
value  of  each. 


PART    II. 


279 


Geometry. 

228.  How  Many  Degrees  m  Each  Angle  of  a  Kegular 

Pentagon  ? 

Fig.l.  Pig.  2.. 


1.  Every   regular   pentagon    may    be    divided   into   

equal  isosceles  triangles. 

2.  The  sum  of  the  angles  of  one  triangle  is  equal  to  

right  angles ;  then  the  sum  of  the  angles  of  5  triangles  is 
equal  to  right  angles. 

3.  But  the  sum  of  the  central  angles  in  figure  2,  (a  +  &  + 

c-\-  d-\-  e)  is  equal  to right  angles ;  then  the  sum  of  all 

the  other  angles  of  the  five  triangles  is  equal  to  10  right 
angles,  less  4  right  angles,  or  6  right  angles  =  540°.  But 
the  angular  space  that  measures  540°,  as  shown  in  figure  2, 
is  made  up  of  10  equal  angles,  so  each  one  of  the  angles  is  1 
tenth  of  540°  or  54°.  Two  of  these  angles,  as  1  and  2, 
make  one  of  the  angles  of  the  pentagon ;  therefore  each  angle 
of  the  pentagon  measures  2  times  54°  or  108°. 

4.  Using  the  protractor  construct  a  regular  pentagon  as 
follows : 

(a)  Draw  two  lines  that  meet  in  a  point,  each  line  being  2  inches 
long  and  the  angular  space  between  them  being  108°. 

(b)  Regarding  the  two  lines  as  two  sides  of  a  regular  pentagon, 
draw  two  more  sides  each  2  inches  long  and  joining  those  already 
drawn  at  an  angle  of  108°. 

(c)  Complete  the  figure  by  drawing  the  fifth  side,  and  prove  your 
work  by  measuring  the  last  line  drawn  ^nd  the  other  two  angles. 


280  COMPLETE    ARITHMETIC. 

229.  Miscellaneous  Review. 

Remembering  that  in  speaking  of  the  per  cent  of  loss  or  gain,  the 
cost  is  the  base  unless  otherwise  specified,  tell  the  per  cent  of  loss  or 
gain  in  each  of  the  following : 

1.  Bought  for  2  and  sold  for  3. 

2.  Bought  for  3  and  sold  for  2. 

3.  Bought  for  4  and  sold  for  5. 

4.  Bought  for  5  and  sold  for  4. 

5.  Bought  for  5  and  sold  for  6. 

6.  Bought  for  6  and  sold  for  5. 

7.  Bought  for  8  and  sold  for  10  ;  for  12. 

8.  Bought  for  8  and  sold  for  14 ;  for  16. 

9.  Bought  for  8  and  sold  for  18  ;  for  20. 

10.  Bought  for  8  and  sold  for  4 ;  for  2. 

11.  Mr.  Parker  sold  goods  at  a  profit  of  25%;  the  amount 
of  his  sales  on  a  certain  day  was  S24.60.  How  much  was 
his  profit  ? 

12.  Mr.  Jewell  sold  goods  at  a  loss  of  25%;  the  amount 
of  his  sales  on  a  certain  day  was  $24.60.  How  much  was 
his  loss. 

13.  By  selling  a  horse  for  $156  there  was  a  loss  to  the 
seller  of  20%.  What  would  have  been  his  gain  per  cent  if 
he  had  sold  the  horse  for  $234  ? 

14.  A  bill  was  made  for  wood  that  was  supposed  to  be  4 
feet  long.  It  was  afterwards  found  to  be  only  46  inches 
long.     What  %  should  be  deducted  from  the  bill  ? 

15.  The  marked  price  on  a  pair  of  boots  was  25%  above 
cost.  If  the  dealer  sells  them  for  25%  less  than  the  marked 
price  will  he  receive  more  or  less  than  the  cost  of  the  boots  ? 

16.  If  by  selling  goods  at  a  profit  of  12%  a  man  gains 
$6.60,  what  was  the  cost  of  the  goods  ? 


PEKCENTAGE. 
230.  Discounting  Bills. 

Many  kinds  of  goods  are  usually  sold  "  on  time  " ;  that  is,  the  buyer 
may  have  30,  60,  or  90  days  in  which  to  pay  for  them.  If  he  pays 
for  such  goods  at  the  time  of  purchase,  or  within  ten  days  from  the 
time  of  purchase,  his  bill  is  "  discounted  "  from  1  ^  to  6  ^,  accord- 
ing to  agTeement ;  that  is,  a  certain  part  of  the  amount  of  the  bill  is 
deducted  from  the  amount. 

Example. 

Mr.  Smith  bought  of  Marshall  Field  &  Co.  a  bill  of  goods 
amounting  to  $350.20.  The  discount  for  immediate  pay- 
ment ("spot  cash")  was  1%.  How  much  must  he  pay  for 
the  goods  ? 

1%  of  $350.20  is  $3.50.     $350.20  -  $3.50  =  $346.70. 

Problems. 
"Figure  the  discounts"  on  the  following  bills: 

1.  Bill  of  $324.37,  discounted  at  2%. 

2.  Bill  of  $276.45,  discounted  at  1%. 

3.  Bill  of  $356.50,  discounted  at  3%. 

4.  Bill  of  $536.50,  discounted  at  6%. 

5.  Bill  of  $561.80,  discounted  at  4%. 

(a)  Find  the  sum  of  the  five  bills  before  they  are  dis- 
counted. 

(b)  Find  the  sum  of  the  discounts. 

(c)  Find  the  sum  of  the  five  bills  after  discounting. 

281 


282  COMPLETE    ARITHMETIC. 

Applications  of  Percentage. 
231.   Discounts  from  List  Price. 

Dealers  in  hardware,  rubber  boots  and  shoes,  belting,  rubber  hose, 
and  many  other  kinds  of  goods,  sell  from  a  list  price  agreed  upon 
by  the  manufacturers.  The  actual  price  is  usually  less  than  the  list 
price.  "  20^  off  "  means  that  the  list  price  is  to  be  discounted  20^. 
"  20  and  10  off "  means  that  the  list  price  is  to  be  discounted  20^^, 
and  what  remains  is  to  be  discounted  10^.  Sometimes  as  many  as 
nine  successive  discounts  are  allowed.  Observe  that  in  computing 
these  the  base  changes  with  each  discount. 

Problems. 

Find  the  actual  cost  of — 

1.  500  ft.  f-inch  gas  pipe  (list,  7^  per  ft.)  at  50  and  10 
off. 

2.  350  ft.  i-inch  gas  pipe  (list,  8^  per  ft.)  at  50  and  10 
off. 

3.  200  ft.  1^-inch  gas  pipe  (list,  26^  per  ft.)  at  55  and 
10  off. 

4.  260  ft.  2 -inch  gas  pipe  (list,  35^'  per  ft.)  at  55  and  10 
off. 

5.  48  ^-in.  elbows  (list,  7^  each)  at  65  and  20  off. 

6.  36  J-in.  elbows  (list,  9^  each)  at  65  and  20  off. 
(a)  Find  the  entire  cost  of  the  six  items. 

7.  Find  the  cost  of  12  pairs  men's  rubber  boots  (list, 
$3.00  per  pair)  at  25  and  10  off.  Find  the  cost  of  the  same 
at  35  off.     Why  are  the  results  unlike  ? 

8.  Which  is  the  lower  price,  50  and  10  off  or  60  off  ? 

9.  Bought  for  40  off  from  list  price  and  sold  for  10  off 
from  list  price.     What  was  my  gain  per  cent  ? 

10.  Bought  for  70  off  from  Hst  and  sold  for  50  and  20 
off  from  list.     Pid  I  lose  or  gain  and  how  msLHj  per  cent  ? 


PART  II.  283 

Applications  of  Percentage. 

232.  Selling  "on  Commission." 

When  goods  are  sold  "  on  commission  "  the  selling  price  is  the  base ; 
that  is,  the  seller  receives  a  certain  per  cent  of  the  selling  price  as 
remuneration  for  services. 

Commission  is  the  sum  paid  an  agent,  or  commission 
merchant,  for  transacting  business. 

Problems. 

1.  At  40%,  what  is  the  commission  for  selling  $275  worth 
of  books  ?  If  the  salesman  sells  and  collects  for  40  %  of  the 
selling  price,  how  much  of  the  $275  will  he  retain  and  how 
much  "pay  over"  to  the  man  for  whom  he  sells  the  books  ? 

2.  While  selling  books  on  a  commission  of  40%  my  com- 
mission amounted  to  $56.  What  was  the  selling  price  of 
the  books  ?  If  I  not  only  sold  but  collected  for  40  %  of  the 
selling  price,  how  much  money  should  I  "pay  over"  to  my 
employer  ? 

3.  A  real  estate  agent  sold  a  house  and  lot  for  $4250.  If 
his  commission  is  5%,  how  much  should  he  receive  for  his 
services  ? 

4.  A  real  estate  agent  sold  a  piece  of  property  upon  which 
his  commission  at  5%  amounted  to  $275.  What  was  the 
selling  price  of  the  property.  How  much  should  the  owner 
receive  for  the  property  after  deducting  the  commission  ? 

5.  A  commissiori  merchant  sold  2140  lbs.  of  butter  at  23^ 
a  pound.  After  deducting  his  commission  of  5%  and 
paying  freight  charges  of  $36.50,  and  storage  charges  of 
$21.40,  how  much  should  he  send  to  the  man  for  whom  he 
made  the  sale  ? 


284  COMPLETE   ARITHMETIC. 

Applications  of  Percentage, 
233.  Taxes. 
A  tax  is  a  sum  of  money  paid  for  public  purposes.  A 
tax  on  property  is  reckoned  at  a  certain  per  cent  of  the 
assessed  value  of  the  property.  The  assessed  value  may  or 
may  not  be  the  real  value.  It  is  often  much  below  the  real 
value. 

Problems. 

1.  Mr.  Hardy  has  a  farm  of  240  acres  which  he  values  at 
$24000.  Its  assessed  value  is  $22  per  acre.  If  his  state  tax 
is  |-%,  his  county  tax  1^%,  his  town  tax  ^%,his  school  tax 
2%,  and  his  special  road  and  bridge  tax  1%,  how  much 
money  must  he  pay  as  taxes  on  his  farm  ? 

2.  The  assessed  value  of  the  taxable  property  of  a  certain 
school  district  is  $176,242.25.  If  the  school  tax  is  2|-% 
and  the  collector  receives  2  %  of  the  amount  collected  as  his 
commission,  and  collects  the  entire  amount  of  the  tax,  how 
much  should  the  district  officers  receive  from  this  source  for 
school  purposes  ? 

3.  The  assessed  value  of  Mr.  Eandall's  property  is  $3400. 
At  the  rate  of  15  mills  on  a  dollar,*  how  much  tax  must  he 
pay? 

4.  The  assessed  value  of  the  property  of  a  district  of  a 
certain  city  is  $250,000.  (a)  What  must  be  the  per  cent  of 
taxation  to  raise  $10,000  ?  (b)  What  will  be  the  net  sum 
realized  for  public  purposes  if  the  collector  is  able  to  collect 
only  95%  of  this  tax  and  he  receives  for  his  services  2%  of 
the  amount  collected  ? 

5.  Mr.  Evans's  tax  is  $35.60  ;  the  rate  of  taxation  is  2|^%. 
What  is  the  assessed  value  of  his  property  ? 

*  "  15  mills  on  a  dollar  "  is  the  same  as  11%. 


TART    II.  285 

Applications  of  Percentage. 

234.  Insurance  is  a  guaranty  by  one  party  to  pay  a  cer- 
tain sum  to  another  party  in  the  event  of  loss  or  damage. 

The  policy  is  the  written  contract  given  by  the  under- 
writer to  the  insured.  The  premium  is  the  sum  paid  for 
insurance. 

Problems. 

1.  A  store  valued  at  $7500  was  insured  for  $5000  for  1 
year.  The  rate  of  insurance  was  2%.  What  was  the  amount 
of  the  premium  ? 

2.  A  stock  of  goods  valued  at  $10000  was  insured  for 
$5000.  A  fire  occurred,  but  part  of  the  goods  were  saved. 
It  was  found  that  the  entire  loss  to  the  owner  of  the  goods 
was  $4750.  (a)  How  much  should  he  receive?  (b)  How 
much  should  he  receive  if  the  loss  were  $5750  ?* 

3.  An  insurance  agent  offers  to  insure  my  farm  buildings 
for  $3500  for  1  year  at  1%,  or  for  5  years  at  3%  ;  the  entire 
premium  in  either  case  to  be  paid  in  advance,  (a)  If  I 
accept  the  first  proposition,  how  much  is  the  premium  to  be 
paid  ?     (b)  How  much  if  I  accept  the  second  ? 

4.  What  is  the  rate  of  insurance  on  the  nearest  store  and 
stock  of  goods  ?  On  farm  property  ?  On  village  or  city 
property  other  than  stores  ?t 

5.  A  large  building  was  insured  in  one  company  for 
$25000,  in  another  company  for  $15000.  It  was  damaged 
by  fire  to  the  extent  of  $12800.  How  much  of  the  damage 
should  each  company  pay  ?  J 

*  In  case  of  total  loss  the  owner  would  receive  $5000.  In  case  of  partial  loss  the 
owner  should  receive  the  full  amount  of  the  loss,  provided  it  does  not  exceed  $5000. 

t  Any  insurance  agent  will  be  willing  to  answer  these  questions  for  you. 

X  The  companies  must  share  the  loss  in  proportion  to  the  amount  of  insurance 
carried  by  them. 


286  COMPLETE    ARITHMETIC. 

235.  Miscellaneous  Problems  in  Applications  of 
Percentage. 

1.  A  dealer  who  had  marked  goods  50%  above  the  cost, 
sold  them  after  deducting  10%  from  the  marked  price.  His 
profit  on  that  sale  was  what  per  cent  of  the  cost  of  the 
goods  ? 

2.  By  selling  a  suit  of  clothes  for  $7.20  I  would  lose  an 
amount  equal  to  1 0  %  of  the  cost.  For  what  must  I  sell  the 
suit  to  gain  a  sum  equal  to  10%  of  the  cost  ? 

3.  I  sold  goods  at  a  loss  of  7%.  My  actual  loss  was 
$3.50.     What  was  the  cost  of  the  goods  ? 

4.  The  real  value  of  a  stock  of  goods  was  $8250.  They 
were  insured  for  $5500.  A  fire  occurred  and  the  salvage 
amounted  to  only  $575.  If  the  insurance  company  promptly 
settles  in  accordance  with  the  above  facts  what  is  the  actual 
loss  to  the  owner  of  the  goods  ? 

5.  If  I  sell  goods  on  a  commission  of  12|-%,  what  must 
bt^  the  amount  of  my  sales  in  order  that  I  may  receive  an 
annual  salary  of  $2500  ? 

6.  A  school  numbers  140  pupils.  The  absence  for  one 
week  was  as  follows  :  Monday,  3  days ;  Tues.,  5  days  ;  Wed., 
4  days  ;  Thurs.,  5  days  ;  Friday,  3  days,  (a)  What  was  the 
per  cent  of  absence  ?  (b)  What  was  the  per  cent  of  attend- 
ance? 

7.  Sold  a  horse  for  $120  and  gained  25%.  What  did  the 
horse  cost  me  ? 

8.  When  the  cost  is  f  of  the  selling  price  what  is  the  gain 
per  cent  ? 

9.  When  the  selling  price  is  f  of  the  cost  what  is  the  loss 
per  cent  ? 


PART  II.  2sr 

Algebra. 
236.  Miscellaneous  Problems. 

1.  In  a  school  there  are  896  pupils.  There  are  three 
times  as  many  boys  as  girls.  How  many  girls  ?  How  many 
boys  ? 

2.  A  man  had  235  sheep.  In  the  second  flock  there  were 
15  more  sheep  than  in  the  first.  In  the  third  flock  there 
were  20  fewer  than  in  the  first.  How  many  sheep  in  each 
flock  ? 

Note — Let  x  =  the  number  in  the  first  flock ;  then  x  -f- 15  =  the 
number  in  the  second,  and  x  —  20,  the  number  in  the  third. 

3.  A  man  owns  three  farms.  In  the  second  there  are  half 
as  many  acres  as  in  the  first.  In  the  third  there  are  twice  as 
many  acres  as  in  the  first.  In  all  there  are  560  acres.  How 
many  acres  in  each  farm  ? 

4.  In  an  apple  and  pear  orchard  containing  296  trees, 
there  were  5  more  than  twice  as  many  apple  trees  as  pear 
trees.     How"  many  of  each  kind  ? 

5.  To  a  number  I  add  one  half  of  itself  and  15,  and  have 
150.     What  is  the  number? 

6.  From  three  times  a  number  I  subtract  -|  of  the  number 

and  5,  and  have  37  remaining.     What  is  the  number  ? 

/2x 
Note. — Let  x  =  the  number ;    then  3a:  —(—-{-  5)  =  37.     On 

^   o 
removing  the  parenthesis,  what  signs  must  be  changed?     See  page; 
207,  IL 

7.  If  to  three  times  a  number  I  add  ^  of  the  number  and 
18,  the  sum  will  be  238.     What  is  the  number  ? 

8.  Two  thirds  of  a  number  is  equal  to  the  number 
decreased  by  56.     What  is  the  number? 


288  COMPLETE   ARITHMETIC. 

Algebra. 
237.  Miscellaneous  Problems. 

1.  A  is  50  years  old.  B  is  20  years  old.  In  how  many 
years  will  A  be  only  twice  as  old  as  B  ? 

Note. — Let  x  =  the  number  of  years ;  then  (20  -\-x)  X  2  =  50  -f  x. 

2.  Find  four  consecutive  numbers  whose  sum  is  150. 

Note. — Let  x  =  the  first ;  then  x  -\-l  =  the  second ;  a;  -|-  2  =  the 
third,  etc. 

3.  Find  three  consecutive  numbers  whose  sum  is  87. 

4.  Two  numbers  have  the  same  ratio  as  2  and  3.  and 
their  sum  is  360.     What  are  the  numbers  ? 

5.  Two  numbers  have  the  same  ratio  as  3  and  4,  and 
their  sum  is  168.     What  are  the  numbers  ? 

6.  Two  numbers  have  the  same  ratio  as  2  and  5,  and 
their  difference  is  87.     What  are  the  numbers  ? 

7.  A  has  $350.  B  has  $220.  How  many  dollars  must 
A  give  to  B  so  that  each  may  have  the  same  sum  ? 

Note. — Let  x  =.  the  number  of  dollars  that  must  be  given  by  A 
to  B ;  then  220  +  a;  =  350  -  x. 

8.  C  has  $560.  D  has  $340.  How  many  dollars  must 
C  give  to  D  so  that  each  may  have  the  same  sum  ? 

9.  E  has  $630.  F  has  $240.  How  many  dollars  must 
E  give  to  F  so  that  E  will  have  exactly  twice  as  many  dollars 
asF? 

10.  The  fourth  and  the  fifth   of  a  certain  number  are 
together  equal  to  279.     What  is  the  number  ? 

11.  The  difference  between  1  fourth  and  1  fifth  of  a  cer- 
tain number  is  28.     What  is  the  number  ? 


PART   II.  289 

Qeometry. 

238.  How  MANY  Degrees  in  each  Angle  of  a  Eegular 
Hexagon  ? 

Fig.  1. 


Fig 

.2. 

/ 

A\ 

^ 

\ 

/^ 

a>? 

^ 

TV 

\ 

f/e 

\^ 

7 

\ 

f/l 

1^9/ 

^ 

1.  Every  regular  hexagon  may  be  divided  into  equal 

isosceles  triangles. 

2.  The  sum  of  the  angles  of  one  triangle  is  equal  to  

right  angles,  then  the  sum  of  the  angles  of  6  triangles  is 
equal  to  right  angles. 

3.  But  the  sum  of  the  central  angles  in  Fig.  2  (a  +  &  +  c 

-i- d  -\-  e  -\- f)  is  equal  to  right  angles ;  then  the  sum 

of  all  the  other  angles  of  the  six  triangles  is  equal  to  12 
right  angles  less  4  right  angles,  or  8  right  angles  =  720°. 
But  the  angular  space  that  measures  720°,  as  shown  in  Fig. 
2,  is  made  up  of  12  equal  angles ;  so  each  one  of  the  angles 
is  one  12th  of  720°,  or  60°.  Two  of  these  angles,  as  1  and 
2,  make  one  of  the  angles  of  the  hexagon ;  therefore  each 
angle  of  the  hexagon  measures  2  times  60°,  or  120°. 

4.  Using  the  protractor,  construct  a  regular  hexagon,  mak- 
ing each  side  2  inches  long. 

Observe  that  since  all  the  angular  space  about  a  point  is  equal  to  4 
right  angles,  or  360°,  and  since  the  space  around  the  central  point  of 
the  hexagon  is  divided  into  6  equal  angles,  each  of  these  angles  is 
an  angle  of  (360°  -v-  6)  60°.  But  each  of  the  other  angles  of  these 
triangles  has  been  shown  to  be  an  angle  of  60° ;  so  each  triangle  is 
equiangular.     Are  the  triangles  equilateral  ? 


290  COMPLETE    ARITHMETIC. 

239.  Miscellaneous  Review. 

1.  A  man  buys  goods  for  $60  and  sells  them  for  $75.  He 
gains  dollars. 

(a)  The  gain  equals  what  part  of  the  cost  ?     What  %  ? 

(b)  The  gain  equals  what  part  of  the  selling  price  ?  What 
per  cent  ? 

(c)  The  cost  equals  what  part  of  the  selling  price  ?  What 
per  cent  ? 

2.  When  the  cost  is  2  thirds  of  the  selluig  price  what  is 
the  per  cent  of  gain  ? 

3.  When  the  selling  price  is  2  thirds  of  the  cost  what  is 
the  per  cent  of  loss  ? 

4.  Bought  for  $200  and  sold  for  $300.  What  was  the  per 
cent  of  gain  ? 

5.  Bought  for  $300  and  sold  for  $200.  What  was  the  per 
cent  of  loss  ? 

6.  A  tax  of  15  mills  on  a  dollar  was  levied  in  a  certain 
town,  the  assessed  value  of  the  taxable  property  being 
$475,250.  If  5%  of  the  tax  is  non-collectable  and  if  the 
collector  is  allowed  2%  of  the  amount  collected,  for  his 
services,  how  much  will  be  realized  from  the  levy  ? 

7.  Which  is  the  greater  discount,  "20  and  10  and  5  off" 
or  "35  off"? 

8.  A  sold  goods  for  B  on  a  commission  of  15%.  His  sales 
for  a  certain  period  amounted  to  $780.  If  the  goods  cost  B 
exactly  $600  was  B's  net  profit  more  or  less  than  10%  ? 

9.  A  offers  rubber  boots  at  "  50  and  20  off " ;  B  offers  them 
at  "  20  and  50  off."  The  quality  and  list  price  bemg  the 
same,  which  offer  shall  I  accept  ? 


INTEEEST. 

J40.  Interest  is  compensation  for  the  use  of  money. 

241.  The  money  for  which  interest  is  paid  is  called  the 
principal. 

242.  The  principal  and  interest  together  are  called  the 
amount. 

Note. — Interest  is  usually  reckoned  in  per  cent,  the  principal 
being  the  base ;  that  is,  the  borrower  pays  for  the  use  of  money  a 
sum  equal  to  a  certain  per  cent  of  the  principal.  When  a  man  loans 
money  "  at  6%  "  he  expects  to  receive  back  the  principal,  and  a  sum 
equal  to  6%  of  the  principal  for  every  year  the  money  is  loaned  and 
at  that  rate  for  fractions  of  years. 

Example. 

Find  the  interest  of  $257  for  two  years  at  6%. 
Operation.  Explanation. 

$2^57 

.12  The  interest  of  any  sum  for  2  years  at  Q%  is  12 

hundredths  of  the  principal.  One  hundredth  of  $257 
is  12.57,  and  12  hundredths  of  $257  is  12  times  $2.57 
or  $30.84. 


5.14 
25.7 


$30.84 

1.  Find  the  interest  of  $242  for  3  yr.  at  7%. 

2.  Find  the  interest  of  $375  for  2  yr.  at  6%. 

3.  Find  the  interest  of  $146  for  1  yr.  at  5%. 

4.  Find  the  interest  of  $274  for  3  yr.  at  5%. 

5.  Find  the  interest  of  $375  for  2  yr.  at  8%. 

6.  Find  the  interest  of  $864  for  3  yr.  at  7%. 
(a)  Find  the  sum  of  the  six  results. 

291 


292  COMPLETE    ARITHMETIC. 

Interest. 

243.  To  Compute  Interest  for  any   Number  of  Years 
AND  Months. 

Note. — The  interest  for  1  month  is  1  twelfth  as  much  as  it  is  for 

1  year ;  for  2  months,  2  tweKths  or  1  sixth,  etc. 

Example. 
Find  the  interest  of  $324.50  for  2  yr.  5  mo.  at  6%. 

Operation  and  Explanation  No.  1. 
Interest  of  1324.50  for  1  year    at  1  ^  =    ^3.245 
Interest  of  ^324.50  for  1  year   at  6^  =  $19,470 
Interest  of  $324.50  for  2  years  at  6  ^  =  $38,940 

Interest  of  $324.50  for  1  mo.     aiQfc^    $1.6225 
Interest  of  $324.50  for  5  mo.     at  6  ^  =  8.1125 

Interest  of  $324.50  for  2  yr.  5  mo.  at  6  ^  =  $47.0525 

Operation  No.  2.  Explanation. 

2  yr.  5  mo.  =  2^^  years.  The  interest  of  any  sum  for  2  yi-. 
2—5    times   06  =   14^             ^  ™^°'  ^^  ^^^'  hundredths  of  the  prin- 

^  ^               ]      ~  '  ^'            cipal. 

$3'24.50  1  hundredth  of  $324.50  is      $3.2450 

•14j-  I  hundredth  of  $324.50  is      $1.6225 

1.6225  4  hundredths  of  $324.50  is    12.9800 

12  9800  10  hundredths  of  $324.50  is    32.450 

32  450  14|  hundredths  of  $324.50  is  $47.0525 


$47.0525 


Problems. 

1.  Find  the  interest  of  $325.40  for  1  yr.  6  mo.  at  7%. 

2.  Find  the  interest  of  $420.38  for  2  yr.  10  mo.  at  6% 

3.  Find  the  interest  of  $221.60  for  2  yr.  3  mo.  at  6%. 

4.  Find  the  interest  of  $145.20  for  1  yr.  9  mo.  at  5%. 

5.  Find  the  interest  of  $340.10  for  3  yr.  1  mo.  at  4%. 
(a)  Find  the  sum  of  the  five  results. 


PART    I.  293 

Interest. 

244.  To  Compute  Interest  for  any  Number  of  Years, 
Months,  and  Days. 

Note. — In  computing  interest,  30  days  is  usually  regarded  as  1 
month,  and  360  days  as  1  year;  so  each  day  is  g^j,  of  a  month  or  gju 
of  a  year. 

Example. 
Find  the  interest  of  $256.20  for  2  yr.  7  mo.  13  days  at  6%. 
Operation  and  Explanation  No.  1. 
Interest  of  $256.20  for  1  yr.     at  1  ^  =    $2.5620 
Interest  of  $256.20  for  1  yi\     at  6  ^  =  $15.3720 
Interest  of  $256.20  for  2  yr.     at  6  ^  =  $30.7440 

Interest  of  $256.20  for  1  mo.    at  6^  =    $1.2810 
Interest  of  $256.20  for  7  mo.   at  6^  =  $8.9670 

Interest  of  $256.20  for  1  da.     at  6^  =      $.04270 
Interest  of  $256.20  for  13  da.  at  6^  =  .5551 

Interest  of  $256.20  for  2  yr.  7  mo.  13  da.  at  6^  =     $40.2661 

Operation  No.  2.  Explanation. 

2  yr.  7  mo.  13  da.  2f  |f  yr. 

2 11  3.  times  .06  =  .15M  The  interest  of  any  sum  tor  2  yr. 

7  mo.  13  da.  at  Q%  is  .15f§  of  the  prin- 
cipal. 


3'  6  0     ^■^^"^'^   • "  "    —  •-^'^60 

$2^56.20 


•l^ef  1  hundredth  of  $256.20  is        $2.5620 


1.8361  |§  of  1  hundredth  of  $250.20  is  $1.8361 

12.8100  5  hundredths  cf  $256.20  is     $12.81 

25.620  10  hundredths  of  $256.20  is  $25.62 


$40.2661  15f^  hundredthsof  $256.20  is$40.2661 

Problems. 

1.  Find  the  interest  of  $350.40  foi  2  yr.  5  mo.  7  da.  at  6%. 

2.  Find  the  interest  of  $145.30  for  1  yr.  7  mo.  10  da.  at  8%. 

3.  Find  the  interest  of  $1 74.20  for  2  yr.  3  mo.  1 5  da.  at  7  %. 
(a)  Find  the  sum  of  the  three  results. 


294  COMPLETE    ARITHMETIC. 

245.    The  '^six'per^cent  method." 

Note.— If  the  teacher  so  prefers,  the  problems  on  the  three  pre- 
ceding pages,  as  well  as  those  that  follow,  may  be  solved  by  the  "  six- 
per-cent  method." 

Explanatory. 

The  interest  at  6%  for  1  yr.  =  .06  of  the  principal. 

The  interest  at  Q%  for  1  mo.  =  ^V  of  .06,  or  .005  of  the  principal. 

The  interest  at  Q%  for  1  day  =  g^^  of  .005,  or  .000^  of  the  principal. 

Reading  Exercise. 

1.  Interest  for  1  yr.  at  6%  = lOOths  of  the  principal; 

for  2  yr.     = lOOths  ;     for  3  yr.     = lOOths  ; 

for  1  mo.  = lOOOths  ;  for  2  mo.  = lOOOths 

for  3  mo.   = lOOOths  ;  for  4  mo.  ^ lOOOths 

for  5  mo.  ■= lOOOths  ;  for  6  mo.  = lOOOths 

for  1  da.    = 1000th  ;    for  3  da.    = 1000th ; 

for  6  da.    = 1000th  ;    for  12  da.  = lOOOths ; 

for  18  da.  = lOOOths  ;  for  24  da.  = lOOOths ; 

for  25  da.  = lOOOths  ;  for  27  da.  = lOOOths. 

Find  the  interest  of  $243.25  for  2  yr.  5  mo.  18  da.  at  6%. 

Operation  and  Explanation. 
Interest  for  2  yr.  =  .12  of  the  principal. 
Interest  for  5  mo.=  .025  of  the  principal. 
Interest  for  18  da.  =  .003  of  the  principal. 
Total  interest  =  .148  of  the  principal. 
1243.25  X  .148  =  $36,001. 

(1)  $36,001  plus  i  of  $36,001  =  int.  of  same  prin.  for  the  same 
time  at  7^. 

(2)  $;36.001  less  I  of  $36,001  =  int.  of  same  prin.  for  the  same 
time  at  5^. 

(3)  How  find  the  interest  at  8^  ?     At  9^?     At  4^?     At  3^? 


PART  II.  295 

Interest. 

246.  Problems   to   be    Solved   by    the   "Six  Per  Cent 

Method." 

1.  Find  the  interest  of  $265  for  1  yr.  3  mo.  13  da.  at  6%.* 

2.  Find  the  interest  of  $346  for  2  yr.  5  mo.  20  da.  at  6%.t 

3.  Find  the  interest  of  $537  for  1  yr.  7  mo.  10  da.  at  6%4 

4.  Find  the  interest  of  $428  for  3  yr.  3  mo.  14  da.  at  6%. 

5.  Find  the  interest  of  $150  for  1  yr.  6  mo.  15  da.  at  6%. 
(a)  Find  the  sum  of  the  five  results. 

6.  Find  the  interest  of  $245.30  for  6  mo.  18  da.  at  7%. 

7.  Find  the  interest  of  $136.25  for  8  mo.  10  da.  at  7%. 

8.  Find  the  interest  of  $321.42  for  5  mo.  15  da.  at  7%. 

9.  Find  the  interest  of  $108.00  for  10  mo.  8  da.  at  7%. 

10.  Find  the  interest  of  $210.80  for  7  mo.  21  da.  at  7%. 

(b)  Find  the  sum  of  the  five  results. 

11.  Find  the  amount  of  $56.25  for  2  yr.  4  mo.  2  da.  at  6%. 

12.  Find  the  amount  of  $31.48  for  1  yr.  5  mo.  11  da.  at  6%. 

13.  Find  the  amount  of  $55.36  for  2  yr.  8  mo.  12  da.  at  6  %. 

14.  Find  the  amount  of  $82.75  for  2  yr.  10  mo.  8  da.  at  6%. 

15.  Find  the  amount  of  $27.35  for  1  yr.  1  mo.  1  da.  at  6%. 

(c)  Find  the  sum  of  the  five  results. 

16.  Find  the  amount  Of  $875  for  3  yr.  8  mo.  15  da.  at  5%. 

17.  Find  the  amount  of  $346  for  2  yr.  6  mo.  12  da.  at  4%. 

18.  Find  the  amount  of  $500  for  1  yr.  7  mo.  18  da.  at  3%. 

19.  Find  the  amount  of  $600  for2  yr.  5  mo.  21  da. at  8%. 

20.  Find  the  amount  of  $825  for  1  yr.  9  mo.  24  da.  at  9%. 

(d)  Find  the  sum  of  the  five  results. 


*lyr. 

.06 

t2yr. 

.12 

nyr. 

.06 

3  mo. 

.015 

5  mo. 

.025 

7  mo. 

.035 

13  da. 

.002^ 

20  da. 

.003J 

10  da. 

.0011 

Total        .077J  Total       .148§  Total       .096f 


296  COMPLETE    ARITHMETIC. 

247.  To  Find  the  Time  Between  Two  Dates. 
Example. 

How  many  years,  months,  and  days  from  Sept.  25,  1892, 
to  June  10,  1896? 

The  Usual  Method. 

1896-6-10  From  the  1896th  yr.,  the  6th  mo.,  and 

1892  -  9  -  25      the  10th  day,  subtract  the  1892nd  yr.,  the 

9th  mo.,  and  the  25th  day.    Regard  a  month 

as  30  days. 

A  Better  Method. 


3-8-15 


From  Sept.  25,  1892,  to  Sept.  25,  1895,  is  3  years. 
From  Sept.  25,  1895,  to  May  25,  1896,  is  8  months. 
From  May  25,  1896,  to  June  10,  1896,  is  16  days. 

Note.— The  two  methods  will  not  alwaj'S  produce  the  same  results.  The  greatest 
possible  variation  is  two  days.  Find  the  time  from  Jan.  22,  1895,  to  March  10, 1897, 
by  each  method,  and  compare  results.  The  difference  arises  from  the  fact  that  the 
month  as  a  measure  of  time  is  a  variable  unit— sometimes  28  days,  sometimes  31. 
The  " usual  method"  regards  each  month  as  30  days ;  the  "  better  method  "  counts  first 
the  whole  years,  then  the  whole  months,  then  the  days  remaining.  By  the  "  usual 
method,"  the  time  from  Feb.  28,  1897,  to  March  1, 1897,  is  3  days ;  by  the  "  better 
method,"  it  is  1  day. 

Problems. 

Find  the  time  by  both  methods  and  compare  the  results. 

1.  From  March  15,  1894,  to  Sept.  10,  1897. 

2.  From  March  15,  1894,  to  Sept.  20,  1897. 

3.  From  May  25,  1895,  to  Sept.  4,  1898. 

4.  From  May  25,  1895,  to  Oct.  4,  1898. 

5.  From  June  28,  1894,.  to  Mch.  1,  1898. 

6.  From  June  28,  1894,  to  May  1,  1898. 

7.  From  Jan.  10,  1892,  to  Jan.  25,  1898. 

8.  From  Jan.  10,  1892,  to  Dec.  25,  1898. 

9.  From  April  15,  1893,  to  Aug.  15,  1898. 


PART  II.  297 

248.  Algebra  Applied  to  Problems  m  Percentage. 

Example. 

75  is  15  per  cent  of  what  number  ? 

Let  X  =  the  number  sought. 

Tlien   or  x,  or         =  75. 

100  100 

Multiplying  by  100,       15  x  =  7500. 
Dividing  by  15,  a?  =  500.     Ans. 

Problems. 

1.  56  is  2  per  cent  of  what  number? 

2.  45  is  5  per  cent  of  what  number  ? 

3.  60  is  12  per  cent  of  what  number  ? 

4.  37  is  4  per  cent  of  what  number  ? 
6.  53  is  8  per  cent  of  what  number  ? 

6.  n  is  r  per  cent  of  what  number  ? 

Let  X  =  the  number  sought. 

Then  =  n 

100 

Multiplying  by  100,          rx  =  100  n 
Dividing  by  r  x  = * 

7.  James  has  $54.20,  and  James's  money  equals  40  per 
cent  of  Henry's  money.     How  much  money  has  Henry  ? 

8.  Mr.  WilHams's  annual  expenses  are  $791.20  ;  this  is 
92  per  cent  of  his  annual  income.  How  much  is  his  annual 
income  ? 

♦Observe  that  every  problem  on  this  page  can  be  solved  by  this  formtUa. 


298  COMPLETE    ARITHMETIC. 

249.  Algebra  Applied  to  Peoblems  in  Percentage. 

Example. 

60  is  what  per  cent  of  75  ? 
Let  X  =  the  per  cent  (number  of  hundredths). 

Then  —  of  75,  or      —=  60. 
100  100 

Multiplying  by  100,     75x  =  6000 

Dividing  by  75,  x  =  80.     Ans. 

Problems. 

1.  180  is  what  per  cent  of  200  ? 

2.  17  is  what  per  cent  of  340  ? 

3.  $87.50  is  what  per  cent  of  $250  ? 

4.  81  is  what  per  cent  of  540  ? 

5.  $75.60  is  what  per  cent  of  $630  ? 

6.  n  is  what  per  cent  of  &  ?  * 


Let 

X  - 

-  the  per  cent. 

Then   —   of  b,  or 
100 

bx 
100 

=  n. 

Multiplying  by  100, 

bx. 

-  IOOtz, 

Dividing  by  b, 

X  - 

IOOti^ 

7, 

Solve  the  first  five  problems  in  four  ways : 

(1)  Let  X  =  the  per  cent  and  solve  as  the  "  example  "  is  solved. 

(2)  Using  one  hundredth  of  each  base  as  a  divisor  and  the  other 
number  mentioned  in  the  problem  as  a  dividend,  find  the  quotient. 

(3)  Find  what  part  the  first  number  mentioned  in  each  problem 
is  of  the  base,  and  change  the  fraction  thus  obtained  to  hundredths. 

(4)  Apply  the  formula, =  x. 

*  The  b  in  problem  6  and  in  the  formula  may  be  thought  oi  as  standing  for  the 
base. 


250. 


PART    II. 
Geometry. 

Some  Interesting  Facts  about  Squares, 
Triangles,  and  Hexagons. 


299 


1.  Four  equal  squares  may  be  so  joined 
as  to  cover  all  the  space  about  a  point. 
Each  angle  whose  vertex  is  at  the  central 
point  of  the  figure  is  an  angle  of  90°. 
Four  times  90°  = degrees. 

2.  Six  equal  equilateral  triangles  may 
be  so  joined  as  to  cover  all  the  space 
about  a  point.  Each  angle  whose  vertex  is 
at  the  central  point  of  the  figure  is  an  angle 
of  60°.     Six  times  60°  = degrees. 

Cut  from  paper  6  equal  equilateral  triangles  and  join  them  as 
shown  in  the  figure. 

3.  Three  equal  hexagons  may  be  so 
joined  as  to  cover  all  the  space  about  a 
point.  Each  angle  whose  vertex  is  at  the 
central  point  of  the  figure  is  an  angle  of 
120°.    Three  times  120°  = degrees. 

Cut  from  paper  3  equal  hexagons  and  join  them  as  shown  in  the 
figure. 

4.  Since  every  regular  hexagon  may 
be  divided  into  six  equal  equilateral  tri- 
angles, as  shown  in  the  figure,  it  follows 
that  the  side  of  a  regular  hexagon  is 
exactly  equal  to  the  radius  of  the  circle 
that  circumscribes  the  hexagon. 


300  COMPLETE   ARITHMETIC. 

251.  Miscellaneous  Reviews. 

1.  Find  the  interest  of  $250  from  Sept.  5,  1896,  to  Jan. 
17,  1898,  at  6%. 

2.  Find  the  amount  of  $340  from  April  19,  1895,  to  Oct. 
1,  1896,  at  6%. 

3.  Find  the  amount  of  $500  from  May  20,  1897,  to  Feb. 
28,  1898,  at  5%. 

4.  Find  the  amount  of  $630  from  July  1, 1896,  to  Nov.  1, 

1896,  at  7%. 

5.  Find  the  amount  of  $800  from  Jan.  1, 1897,  to  Jan.  25, 

1897,  at  6%. 

6.  If  a  60-day  bill  of  $400  is  discounted  2%  for  cash, 
how  much  ready  money  vi^ill  be  required  to  pay  the  bill  ? 

7.  Find  the  amount  of  $392  for  60  days  (two  months) 
at  6%. 

Note. — Observe  that  8392  is  the  answer  to  problem  6.  The 
result  of  problem  7,  then,  is  the  amount  that  the  goods  mentioned 
in  problem  6  would  cost  at  the  end  of  60  days  if  the  purchaser  bor- 
rowed the  money  at  Q%  to  pay  for  them.  Compare  this  result  with 
$400.  How  much  does  the  purchaser  of  the  bill  save  by  borrowing 
the  money  at  6%  to  pay  the  bill  instead  of  letting  the  bill  run  60 
days  and  then  paying  $400  ? 

8.  Find  the  cost  of  goods,  the  list  price  being  $46,  and 
the  discounts  "50  and  10  off'  and  2  off  for  cash." 

9.  My  remuneration  for  selling  goods  on  a  commission  of 
40%  amounted  to  $56.20.  How  much  should  the  man  for 
whom  the  goods  were  sold  receive  ? 

Note.— $56.20  is  40^  of  the  selling  price  of  the  goods.  The  man 
for  whom  the  goods  were  sold  should  receive  60^  of  the  selling  price. 
When  goods  are  sold  on  a  commission  of  40^,  what  the  agent 
receives  equals  what  part  of  what  the  employer  receives?  What  the 
employer  receives  is  how  many  times  what  the  agent  receives  ? 


PEOMISSOEY   NOTES. 

252.  When  a  man  borrows  money  at  a  bank  he  gives  his 
note  for  a  specified  sum  to  be  paid  at  a  specified  time,  and 
he  receives  therefor,  not  the  sum  named  in  the  note,  but 
that  sum  less  the  interest  upon  it  from  the  time  the  note  is 
given  to  the  bank  officials  to  the  time  the  note  is  due. 

253.  The  acceptance  of  a  note  by  the  bank  officials  and 
the  payment  of  a  sum  less  than  it  will  be  worth  at  maturity 
is  called  "  discounting  the  note!' 

254.  The  proceeds  of  a  note  is  the  sum  paid  for  it.  As  a 
rule,  the  notes  discounted  at  a  bank  are  those  which  bear  no 
interest,  and  the  date  of  discounting  is  usually  the  same  as 
the  date  of  the  note,  though  not  necessarily  so. 

Example. 

Find  the  discount  and  proceeds  of  the  following: 

S800.  Jacksonville,  III.,  Jan.  10,  1898. 

Sixty  days  after  date  I  promise  to  pay  James  Kice,  or 
order,  eight  hundred  dollars,  value  received. 

Arthur  Williams. 
Discounted  at  6%,  Jan.  10,  1898. 

Erom  the  date  of  discount  to  the  date  of  maturity  it  is 
60  days.     Interest  of  $800  for  60  days  =  $8.00. 
Proceeds  of  note  =  $800  ~  $8  : 


Observe  that  the  bank  receives  the  interest  on  $800  for  two  months 
at  Q%  for  the  use  of  $792  for  two  months.  The  actual  rate  of  interest 
is  therefore  a  Httle  more  than  the  rate  named. 

301 


302  COMPLETE    ARITHMETIC. 

255.  Problems  in  Bank  Discount. 

1.  $375.00.  Chicago,  III.,  Apr.  5,  1898. 

Thirty  days  after  date  I  promise  to  pay  to  the  order 
of  John  Smith,  three  hundred  seventy-five  dollars,  at  the 
Union  National  Bank.    Value  received.         James  White. 
Discounted  April  5,  at  6  %.     Find  proceeds. 

2.  $450.00.  Aurora,  III.,  March  10,  1898. 

Sixty  days  after  date  I  promise  to  pay  to  Wm. 
George,  or  order,  four  hundred  fifty  dollars,  at  the  Old  Second 
National  Bank.    Value  received.  F.  D.  Winslow. 

Discounted  March  10,  at  7  %.     Find  proceeds. 

3.  $2300.00.  Waukegan,  III.,  Feb.  9,  1898. 

Ninety  days  after  date  I  promise  to  pay  to  the  order 
of  John  Mulhall,  two  thousand  three  hundred  dollars,  at  the 
Security  Savings  Bank.     Value  received. 

Chas.  Whitney. 

Discounted  Feb.  9,  at  7  ^.     Find  proceeds. 

4.  $5000.00.  Serena,  III.,  Jan.  20,  1898. 

Sixty  days  after  date  I  promise  to  pay  to  John  Parr, 
or  order,  five  thousand  dollars.     Value  received. 

Harry  Brown. 

Discounted  Jan.  20,  at  7  %.     Find  proceeds. 

5.  $3500.  Boston,  Mass.,  April  12,  1898. 

Sixty  days  after  date  I  promise  to  pay  to  W.  J.  But- 
ton, or  order,  three  thousand  five  hundred  dollars.  Value 
received.  Harry  Wilson. 

Discounted  Apr.  24,  at  6  %.    Find  proceeds. 


PART   II.  303 

Promissory  Notes. 
256.  The  Discounting  of  Interest-Bearing  Notes. 

Whenever  a  note  is  presented  at  a  bank  to  be  discounted,  its  value 
at  maturity  is  regarded  as  the  sum  upon  which  the  discount  is  to  be 
reckoned.  When  the  note  is  not  interest-bearing,  its  face  value  is 
its  value  at  maturity.  In  the  discounting  of  short-time  notes  at 
banks  it  is  customary  to  find  the  exact  number  of  days  from  the 
date  of  discounting  to  the  date  of  maturity  and  to  regard  each  day 
as  ^l^  of  a  year. 

Example. 

$750.00.  Austin,  III.,  Jan.  1,  1898. 

Six  months  after  date,  I  promise  to  pay  to  the  order 
of  N.  D.  Gilbert,  seven  hundred  fifty  dollars,  with  interest  at 
the  rate  of  six  per  cent  per  annum.  0.  T.  Bright. 

Discounted  at  a  bank,  May  10,  1898,  at  7%. 

Value  of  the  note  at  maturity,     $772.50 

Discount,  52  days  at  7%,  7.81 

Proceeds,  $764.69 

1.  $540.00.  ToPEKA,  Kansas,  Dec.  10,  1897. 

Five  months  after  date,  I  promise  to  pay  to  the 
order  of  J.  C.  Thomas,  five  hundred  forty  dollars,  with  interest 
at  the  rate  of  six  per  cent  per  annum.         Hiram  Baker. 
Discounted  at  a  bank,  April  1,  1898,  at  7^. 

2.  $325.00.  Earlville,  III.,  Feb.  1,  1898. 

Six  months  after  date,  I  promise  to  pay  to  the  order 
of  Wm.  E.  Haight,  three  hundred  twenty-five  dollars,  with 
interest  at  the  rate  of  six  per  cent  per  annum. 

Chas.  Hoss 
Discounted  at  a  bank,  June  1,  1898. 


304  COMPLETE    ARITHMETIC. 

257.  Partial  Payments  on  Notes. 

A  partial  payment  is  a  part  payment  made  upon  a  note 
before  the  time  of  final  settlement. 

There  are  two  methods  in  common  use  of  finding  the  value  of  a 
note  at  maturity,  upon  which  one  or  more  partial  payments  have 
been  made.  The  first  method  is  often  applied  to  computation  of 
"  short-time  notes,"  such  as  run  one  year  or  less.  A  formal  state- 
ment of  this  method  is  called  the — 

Merchants'  Eule. — Find  the  amount  of  the  face  of  the 
note  from  the  date  to  mat^irity.  Then  find  the  amount  of 
each  payment  from  the  time  it  was  made  to  the  maturity  of 
the  note.  From  the  amount  of  the  face  of  the  note  subtract 
the  sum  of  the  amounts  of  the  payments. 

$450.  Waukegan,  III.,  Jan.  1,  1897. 

One  year  after  date,  I  promise  to  pay  to  the  order  of 
Wm.  E.  Toll,  four  hundred  fifty  dollars,  with  interest  at  six 
per  cent.     Value  received.  Jerome  Biddlecom. 

Part  payments  were  made  upon  this  note  as  follows;  March  1, 
1897,  $250;  May  1,  1897,  |150. 

Amount  of  the  face  of  the  note  for  one  year,  $477.00 

Amount  of  $250,  March  1,  '97,  to  Jan  1,  '98,    $262.50 
Amount  of  $150,  May  1,  '97,  to  Jan.  1,  '98,        156.00    $418.50 
Value  at  maturity,  $  58.50 

1.     $1000.00.  Springfield,  III.,  Jan.  1,  1898. 

Eight  months  after  date,  I  promise  to  pay  to  the 
order  of  Fred  H.  Wines,  one  thousand  dollars,  with  interest 
at  six  per  cent.     Value  received.  J.  H.  Freeman. 

Part  payments  were  made  upon  this  note  as  follows :  Apr.  1, 1898, 
$350;  June  15,  1898,  $240. 

How  much  was  due  on  this  note  at  maturity,  Sept.  1,  1898? 


PART   II.  305 

Promissory  Notes. 

258.  A  decision  of  the  United  States  Supreme  Court  many 
years  ago  led  to  the  very  general  adoption  in  the  solution  of 
"  partial  payment  problems,"  of  what  is  now  known  as  the 
United  States  Eule. — Find  the  amount  of  the  principal  at 
the  time  of  the  first  payment.  Subtract  the  first  payment 
from  this  amount.  The  remainder  is  a  new  principal,  upon 
which  find  the  amount  at  the  time  of  the  second  payment.  Sub- 
tract the  second  payment  from  this  amount.  Continue  this  pro- 
cess to  the  time  of  settlement.     The  last  amount  is  the  sum  due. 

An  exception  to  the  foregoing  rule  is  required  when  there  is  a  pay- 
ment which  is  less  than  the  interest  due  at  the  time  the  payment  is 
made.  In  such  case  the  payment  is  treated  as  though  made  at  the 
time  of  the  next  payment ;  and  if  the  two  payments  together  do  not 
equal  the  interest  due,  the  sum  of  both  is  again  carried  forward. 

$950.  Petersburg,  III.,  Jan.  1,  1895.    « 

On  or  before  Jan.  1,  1898, 1  promise  to  pay  to  N.  W. 

Branson,  or  order,  nine  hundred  fifty  dollars,  with  interest 

at  six  per  cent.     Value  received.  B.  Laning. 

Part  payments  were  made  on  this  note  as  follows:  July  1,  1895, 

$150;  Jan.  1,  1896,  $200;    Jan.  1,  1897,  $250.     Find  the  amount 

due  Jan.  1,  1898. 

Amount  of  $950,  July  1,  1895  (6  mo.),      $978.50 
Subtract  first  payment,  150.00 

New  principal,  $828.50 

Amount  of  $828.50,  Jan.  1,  1896  (6  mo.),  $853.35 
Subtract  second  payment,  200.00 

New  principal,  $653.35 

Amount  of  $653.35,  Jan.  1,  1897  (1  year),  $692.55 
Subtract  third  payment,  250.00 

New  principal,  $442.55 

Amount  of  $442.55,  Jan.  1, 1898  (1  year),  $469.10 


306  complete  arithmetic. 

259.  Problems  in  "Partial  Payments." 

1.  Date  of  note,  Jan.  1,  1896. 
Pace  of  note,  $800. 

Kate  of  interest,  six  per  cent. 
Payment  May  1,  1896,  $125. 
Payment  Dec.  1,  1896,  $230. 
Find  amount  due  Jan.  1,  1898. 

2.  Date  of  note,  July  10,  1896. 
Face  of  note,  $500. 

Rate  of  interest,  six  per  cent. 
Payment  Dec.  22,  1896,  $200. 
Payment  July  15,  1897,  $200. 
Find  amount  due  Jan.  1,  1898. 

'     3.  Date  of  note,  Jan.  1,  1897. 
Face  of  note,  $600. 
Eate  of  interest,  six  per  cent. 
Payment  Sept.  1,  1897,  $100. 
Find  amount  due  Jan.  1,  1898. 

Note. — Solve  problem  3  by  the  Merchants'  Rule  and  by  tha 
United  States  Rule.  The  answer  obtained  by  the  latter  will  be  48 
cents  larger  than  the  answer  obtained  by  the  former. 

Observe  that  the  Merchants'  Rule  is  based  upon  the  supposition 
that  interest  is  not  due  until  the  time  of  settlement  of  the  note, 
while  the  U.  S.  Rule  is  based  upon  the  supposition  that  interest  is 
due  whenever  a  payment  is  made.  By  applying  the  latter  rule  to 
problem  3,  $24  of  interest  must  be  paid  Sept.  1 ;  by  applying  the 
former  rule  to  the  same  problem  the  entire  $100  paid  Sept.  1  applies 
in  payment  of  principal.  The  answer,  then,  by  the  U.  S.  Rule  must 
be  greater  than  the  answer  by  the  Merchants'  Rule,  by  the  interest 
on  |24  for  four  months,  or  48  cents. 


PART  II.  307 

Algebra. 
260.   Algebra  Applied  to  Some  Problems  in  Interest 

Example. 

What  principal  at  6%  will  gain  $96  in  2  yrs.? 

Let  X  =  the  principal. 
Since  the  interest  at  6%  for  2  years  equals  -^^  of  the 
principal 

then  11.,  or  1^  =  96. 
100  100 

Multiplying  by  100  12ic  =  9600 

X  =  800.* 

Problems. 

1.  What  principal  at  6%  will  gain  $67.50  in  2  years  6 
months  ? 

2.  What  principal  at  6%  will  gain  $27.20  in  1  year  4 
months  ? 

3.  What  principal  at  7%  will  gain  $87.50  in  2  years  6 
months  ? 

4.  What  principal  at  5%  will  gain  $187.50  in  five  years  ? 

5.  What  principal  at  8%  will  gain  $64  in  1  year  3  months  ? 

6.  What  principal  at  6%  will  gain  $61.20  in  2  years  6 
months  1 8  days  ? 

153  153a; 

Let  X  =  the  principal,  then x,  or =  $61.20. 

^        ^  1000  1000 

(a)  Find  the  sum  of  the  six  answers. 

*  What  principal  at  a%  will  gain  6  dollars  in  c  years? 


308  COMPLETE    ARITHMETIC. 

261.  Algebra  Applied  to  Some  Problems  in  Interest. 

Example. 
What  principal  at  6%  will  amount  to  $828.80  in  2  years  ? 

Let  X  —  the  principal. 

1  2r 
Then    x^  J_:=  828.80. 
100 

Multiplying  by  100,     100^  +  12^  =  82880. 

Uniting,  112.^?  =  82880 

X  =  740.* 

Problems. 

1.  What  principal  at  6%  will  amount  to  $368  in  2  years 
6  months  ?  f 

2.  What  principal  at  6%  will  amount  to  $588.30  in  1  year 
10  months? 

3.  What  principal  at  5%   will  amount  to  $393.75  in  2 
years  6  months  ? 

4.  What  principal  at  5%  will  amount  to  $287.50  in  three 
years  ? 

5.  What  principal  at  6%  will  amount  to  $458.80  in  2 
years  5  months  and  12  days? 

147  14:7x 

Let  X  -  the  principal ;  then  -r-rr^>  or  Yo^(\  ~  ^^^  interest, 

147a; 

and  X  A =  the  amount,  $458.80. 

1000 

fa)  Find  the  sum  of  the  five  answers. 

*  What  principal  at  ai  will  amount  to  6  dollars  in  c  years? 
f  To  THE  Pupil.— Prove  each  answer  obtained  by  finding  its  amount  for  the 
given  time  at  the  given  rate 


IPAKT   It. 


309 


Geometry. 
262.  The  Area  of  a  Eectangle. 

1.  One  side  of  every  rectangle 
may  be  regarded  as  its  base.  The 
side  perpendicular  to  its  base  is  its 
altitude. 

2.  The  number  of  square  units  in 
the  row  of  square  units  next  to  the 
base  of  a  rectangle,  taken  as  many 

times  as  there  are  linear  units  in  its  altitude,  equals  the 
number  of  square  units  in  its  area.  In  the  figure  given,  we 
have  4  sq.  units  x  3  =  12  sq.  units. 

Note  1. — In  the  above,  it  is  assumed  that  the  base  and  altitude 
are  measured  by  the  same  linear  unit,  and  that  the  square  unit  takes 
its  name  from  the  linear  unit. 

Note  2. — In.  the  actual  finding  of  the  area  of  rectangles  for  prac- 
tical purposes,  the  work  is  done  mainly  with  abstract  numbers  and 
the  proper  interpretation  is  given  to  the  result.  There  can  be  no 
serious  objection  to  the  rule  for  finding  the  area  of  rectangles  as 
given  in  the  old  books,  provided  the  pupil  is  able  to  interpret  it. 

Rule. — To  find  the  area  of  a  rectangle,  "  multiply  its  base  hy  its 
altitude.'' 

Problems. 

1.  Find  the  area  of  the  surface  of  a  cubical  block  whose 
edge  is  9  inches  in  length. 

2.  Find  the  area  in  square  yards  of  a  rectangular  piece 
of  ground  that  is  36  feet  by  45  feet. 

3.  Find  the  area  in  acres  of  a  rectangular  piece  of  land 
that  is  92  rods  by  16  rods. 

4.  Find  the  area  in  square  rods  of  a  piece  of  ground  that 
is  99  feet  by  66  feet. 


310  COMPLETE    ARITHMETIC. 

263.  Miscellaneous  Review. 

1.  Clarence  Marshall  wished  to  borrow  some  money  at  a 
bank.  He  was  told  by  the  president  of  the  bank  that  they 
(the  bank  officials)  were  "discounting  good  30-day  paper" 
at  7%.  Mr.  Marshall's  name  being  regarded  as  "good," 
he  drew  his  note  upon  one  of  the  forms  in  use  at  the  bank, 
for  $1000  payable  in  thirty  days  without  interest.  On  the 
presentation  of  this  note  to  the  cashier,  how  much  money 
should  he  receive  ? 

2.  If  Mr.  Marshall's  note  described  in  problem  1  was 
dated  April  10,  1898,  (a)  when  must  it  be  paid?  (b)  How 
much  money  will  he  pay  when  he  "  takes  up "  the  note  ? 
(c)  Does  he  pay  for  the  use  of  the  money  borrowed,  at  the 
rate  of  exactly  7  %  per  annum  ? 

3.  If  a  bank  is  discounting  at  7%,  how  much  should  be 
given  for  a  note  of  $200  due  in  two  months  from  the  time  it 
is  discounted  and  bearing  interest  for  the  two  months  at 
the  rate  of  6  %  per  annum  ? 

4.  Find  the  value  at  the  time  of  settlement  of  the  follow- 
ing note : 

Date  of  note,  Apr.  1,  1896. 
Face  of  note, $300.  Eate,  6%. 
Payment,  Aug.  1,  1896,  $75. 
Payment,  Apr.  1,  1897,  $80. 
Settled,  Aug.  1,  1898. 

5.  What  principal  at  6%  will  gain  $6  in  1  year  4  months  ? 

6.  What  principal  at  6%  will  amount  to  $81  in  1  year 
4  months  ? 

7.  Find  the  area,  in  square  feet,  of  a  walk  4  feet  wide 
around  a  rectangular  flower-bed  that  is  40  feet  long  and  12 
feet  wide. 


STOCKS   AND   BONDS. 

264.  Some  kinds  of  business  require  so  much  capital  that 
many  persons  combine  to  provide  the  necessary  money.  Such 
a  combination  of  men  organized  under  the  laws  of  a  State,  the 
capital  being  divided  into  shares,  is  known  as  a  corporation, 
or  stock-company.  Those  who  own  the  shares  are  called 
stock-holders.  The  stock-holders  elect  from  their  own 
number  certain  men  to  manage  the  business.  These  man- 
agers are  called  directors. 

265.  The  nominal  value  of  a  share  is  its  face  value ;  that 
is,  the  sum  named  on  its  face.  Large  corporations,  usually, 
though  not  always,  divide  their  capital  into  $100  shares. 

If  the  business  is  prosperous,  shares  may  sell  on  the  market  for 
more  than  their  nominal  value.  The  stock  is  then  said  to  be  "  above 
par,"  or  "  at  a  premium.'^ 

If  the  business  does  not  prosper,  the  shares  may  sell  on  the  mar- 
ket for  less  than  their  nominal  value.  The  stock  is  then  said  to  be 
"  helow  par,*'  or  "  at  a  discount." 

266.  If  the  business  is  profitable,  a  part  or  all  of  the  earn- 
ings is  periodically  divided  among  the  stock-holders.  The 
sum  divided  is  called  a  dividend. 

Dividends  are  always  reckoned  on  the  nominal  or  par  value  of 
the  stock.  If  a  corporation  declares  a  2%  dividend,  it  pays  to  each 
stock-holder  a  sum  equal  to  2%  of  the  nominal  value  of  the  stock 
which  he  owns. 

267.  The  kinds  of  business  which  are  usually  conducted 
by  corporations,  are:  The  mining  of  coal,  silver,  gold,  etc.; 
the  operation  of  gas  works,  railroads,  large  manufacturing 
establishments  of  all  kinds,  creameries,  etc. 

3U 


312 


COMPLETE    ARITHMETIC. 
Certificate  of  Stock. 


/  /: 


1.  Examine  the  above  certificate.  What  part  of  the  entire 
stock  of  the  Werner  School  Book  Company  would  the  owner 
of  one  hundred  shares  have  ? 

2.  A  3%  dividend  would  require  how  much  money  from 
the  treasury  of  the  Company  ?  How  much  money  should 
the  owner  of  one  hundred  shares  receive  ? 

3.  If  a  4%  dividend  is  declared  how  much  money  should 
the  owner  of  seven  hundred  and  fifty  shares  of  stock  receive 
as  his  share  of  the  dividend  ? 


PART    II.  313 

Stocks. 

268.  History  of  a  Stock  Company. 

The  farmers  of  a  certain  community  agreed  to  combine  in 
the  building  and  management  of  a  creamery.  It  was  deter- 
mined that  a  capital  of  $5000  was  necessary.  This  was 
divided  into  50  shares  of  $100  each.  Men  then  came  for- 
ward and  contributed  as  follows  • 


A  took  5  shares  and  paid  in 
B  took  3  shares  and  paid  in 
C  took  6  shares  and  paid  in 
D  took  4  shares  and  paid  in 
E  took  7  shares  and  paid  in  $700. 
F  took  2  shares  and  paid  in  $200. 
G  took  10  shares  and  paid  in 
H,  T,  J,  K,  L,  M,  N,  O,  P,  Q,  R,  S,  and  T  took  1  share  each  and 
paid  in  $100  each. 

(a)  At  the  end  of  the  first  year  the  directors  declared  an 
8%  dividend.     How  much  did  A  receive  ?  B  ?  C  ?  K  ? 

(b)  Wliat  was  the  entire  amount  of  the  money  divided 
among  the  stock-holders  at  the  end  of  the  first  year  ? 

(c)  Soon  after  this  dividend  was  paid,  A  sold  his  stock  to 
X  at  a  premium  of  10%.  How  much  did  A  receive  for  his 
stock  ? 

(d)  At  the  end  of  the  second  year  the  profits  were  found 
to  be  comparatively  small,  and  the  directors  could  pay  a  div- 
idend of  only  3%.     How  much  did  X  receive  ?    B  ?   C  ?   K  ? 

(e)  What  was  the  entire  amount  of  the  money  divided 
among  the  stock-holders  at  the  end  of  the  second  year  ? 

(f)  Soon  after  this  dividend  was  paid,  B,  C,  D,  E,  and  F 
sold  their  stock  to  Y  at  a  discount  of  10%.  How  much  did 
this  stock  cost  y  ? 


314  COMPLETE    ARITHMETIC. 

History  of  a  Stock  Company — Continued. 

(g)  At  the  end  of  the  third  year,  the  profits  were  so  small 
that  no  dividend  was  declared.  The  stock-holders  became 
disheartened  and  many  of  them  offered  to  sell  their  stock  at 
a  large  discount.  Z  appeared  in  the  market  and  bought  at 
"50^  on  the  dollar"  all  the  stock  of  the  company  except 
that  owned  by  X  and  Y.     How  much  did  this  stock  cost  Z? 

(h)  At  the  end  of  the  fourth  year,  the  directors,  X,  Y, 
and  Z,  declared  a  10%  dividend.  How  much  money  was 
divided  and  how  much  did  each  receive  ? 

(i)  Before  the  close  of  the  fifth  year,  the  property  burned 
and  the  lot  upon  which  it  stood  was  sold.  After  the  insur- 
ance money  had  been  received,  the  book  accounts  collected, 
and  all  debts  paid,  there  remained  in  the  treasury  of  the 
company  $4350.  How  much  of  this  money  should  each 
stock-holder,  X,  Y,  and  Z,  receive  ? 

(j)  Did  this  creamery  enterprise  prove  a  good  investment 
forX?     ForY?     ForZ?     For  A?     For  B?.  For  M? 

Miscellaneous  Problems. 

1.  The  directors  of  a  company  whose  capital  is  $50000 
determined  to  distribute  among  the  stock-holders  $2500  of 
profits,  (a)  A  dividend  of  what  per  cent  shall  be  declared  ? 
(b)  How  much  will  a  man  receive  who  owns  15  lOO-doUar 
shares  ? 

2.  A  company  whose  capital  is  $75000  pays  a  dividend  of 
3%.  (a)  How  much  money  is  divided  among  the  stock- 
holders ? 

3.  Mr.  Steele  owns  20  shares  ($100)  in  the  C,  B. &  Q.  R R. 
He  receives  as  his  part  of  a  certain  dividend  $110.  What  is 
the  per  cent  of  the  dividend  ? 


PART    II.  315 

Bonds. 

269.  A  bond  is  a  very  formal  promissory  note  given  by  a 
government  or  other  corporate  body,  as  a  railway  or  a  gas 
company,  for  money  borrowed.  Bonds  usually  have  attached 
to  them  small  certificates  called  coupons.  These  are  really 
little  notes  for  the  interest  that  will  be  due  at  different  times 
Thus,  a  10-year  bond  for  $1000  with  interest  at  6%  payable 
semi-annually  may  have  20  coupons  attached,  each  calling 
for  $30  of  interest. 

270.  Money  invested  in  bonds  yields  a  specified  income ; 
but  the  income  from  money  invested  in  stocks  depends  upon 
the  profits  of  the  company. 

271.  Bonds,  like  stocks,  are  sometimes  sold  for  more  than 
their  face  value.  They  are  then  said  to  be  "  above  par"  or 
"  at  a  premium."  Like  stocks,  too,  they  are  sometimes 
" helow par"  or  "  at  a  discount." 

1.  What  is  the  semi-annual  interest  on  two  lOOO-doUar 
U.S.  5%  bonds? 

2.  What  sum  should  be  named  on  each  coupon  of  a  1000- 
doUar  city  bond  if  the  interest  is  payable  annually  at  the 
rate  of  7%  ? 

3.  To  raise  the  money  to  build  a  court-house,  a  certain 
county  issued  $50000  worth  of  6%  ten-year  bonds.  These 
sold  upon  the  market  at  2%  premium,  (a)  How  much 
money  was  received  for  the  bonds  ?  (b)  How  much  did  A 
pay,  who  bought  three  lOOO-doUar  bonds?  (c)  If  the  in- 
terest was  payable  semi-annually,  how  much  should  A 
receive  each  6  months  on  this  investment  ? 

4.  Has  the  county  or  city  in  which  you  live  any  "bonded 
indebtedness  "  ?  If  so,  how  much,  and  what  is  the  rate  of 
interest  ? 


316 


COMPLETE    ARITHMETIC. 
City  Bond  with  Coupons  Attached. 


■ro 


oi'WKcmoi 


Mixsrm/ioma 


Note. — Bonds  are  made  in  great  variety  both  as  to  form  and 
content;  but  in  all,  indebtedness  is  acknowledged,  and  the  amount, 
rate  of  interest,  and  time  of  payment  for  both  principal  and  interest, 
named.  The  above  is  a  very  short  and  concise  form  of  Bond  (much 
reduced  in  size)  and  is  an  exact  copy  of  one  prepared  for  actual  use. 

1.  Examine  the  above  Bond.  If  the  time  it  is  to  run  is 
five  years,  how  many  coupons  should  be  attached? 

2.  If  the  Bond  is  dated  Jan.  1,  1898,  what  date  should 
be  written  in  each  coupon? 

3.  If  the  face  of  the  Bond  is  $100  and  the  rate  5%,  what 
sum  should  be  written  in  each  coupon  ? 


PART    II.  317 

272.  Algebra  Applied  to  Some  Problems  m  Interest. 

Example. 

At  what  rate  per  cent  will  $500  gain  $55  in  2  yrs.?* 
Let  X  =  the  rate, 

then  — —  of  500,  or  ,  or  10^  =  the  interest, 

and  lOeZ?  =  55 

Dividing  x  =  6^- 

Problems. 

1.  At  what  rate  per  cent  will  $450  gain  $72  in  2  years? 

2.  At  what  rate  per  cent  will  $320  gain  $48  in  3  years  ? 

3.  At  what  rate  per  cent  will  $560  gain  $84  in  2  years 
6  months  ? 

4.  At  what  rate  per  cent  will  $600  gain  $75  in  2  years 
6  months  ? 

5.  At  what  rate  per  cent  will  $600  gain  $114  in  2  years  4 
months  15  days  ? 

2  yr.  4  mo.  15  da.  =  2|-  years. 
Let  X  =  the  rate.     Then  ^  of  600  =  114. 

Note. — Problem  5  may  be  solved  arithmetically  by  finding  the 
interest  of  $600  for  2  yr.  4  mo.  15  da.  at  Q%.  Divide  this  interest 
by  6  (to  find  the  interest  at  1%)  and  find  how  many  times  the  quo- 
tient is  contained  in  |114. 

♦  The  arithmetical  solution  of  this  problem  is  as  follows :  The  interest  of  $500  for 
2  years  at  1^  is  igj,  of  $500.  ih  of  S500  =  $10.  To  gain  $55  in. 2  years,  $500  must  be 
loaned  at  as  many  per  cent  as  $10  is  contained  times  in  $55.  It  is  contained  5| 
times ;  so  $500  must  be  loaned  at  5i  %  to  gain  $55  in  2  years.  Observe  that  by  this 
method  we  divide  the  given  interest  by  the  interest  of  the  principal  for  the  given  time  at 
one  per  cent. 


318  COMPLETE    ARITHMETIC. 

Algebra. 

273.   Algebra  Applied  to  Some  Problems  in  Interest. 

Example. 
In  how  long  a  time  will  $650  gain  $97.50  at  6%? 
Let  X  —  the  number  of  years, 

then  —  of  650  =  97.50 

100 

Simplifying,  39^  =  97.50 

Dividing,  ^  =  2.5* 

Problems. 

1.  In  how  long  a  time  will  $400  gain  $30  at  5%? 

2.  In  how  long  a  time  will  $600  gain  $96  at  6%? 

3.  In  how  long  a  time  will  $800  gain  $68  at  6%? 

4.  In  how  long  a  time  will  $500  gain  $56  at  6%? 

5.  In  how  long  a  time  will  $400  gain  $29  at  6%? 

Review  Problems. 

6.  What  principal  at  8%  will  gain  $124.80  in  3  years? 

7.  What  principal  at  7%  will  amount  to  $410.40  in  2 
years  ? 

8.  At  what  rate  per  cent  will  $900  gain  $72  in  2  years? 

9.  In  how  long  a  time  will  $1000  gain  $160  at  6  per  cent? 
10.  What  p/incipal  at  5%  will  amount  to  $736  in  3  years  ? 

To  the  Pupil. — Prove  each  answer  by  finding  the  interest  on 
the  given  principal  at  the  given  rate  for  the  time  obtained. 

*  The  arithmetical  solution  of  this  problem  is  as  follows :  The  interest  of  $650  for 
one  year  at  %  is  839.  As  many  years  will  be  required  to  gain  $97.50  as  $39.00  is  con- 
tained times  in  $97.50.  It  is  contained  2\  times  ;  so  in  2J  years  $650  will  gain  $97.50. 
Observe  that  by  either  method  we  divide  the  given  interest  by  the  interest  of  the  principal 
for  1  year  at  the  aiven  rate. 


PART    II. 


319 


Geometry. 

274.  The  Akea  of  a  Ehomboid  * 

1.  One  side  of  a  rhomboid  may- 
be regarded  as  its  hase.  The  per- 
pendicular distance  from  the  base 
to  the  opposite  side  is  its  altitude. 

2.  Convince  yourself  by  measurements  and  by  paper- 
cutting  that  from  every  rhomboid  there  may  be  cut  a  tri- 
angle, (abc),  which  when  placed  upon  the  opposite  side,  {def), 
converts  the  rhomboid  into  a  rectangle  (adeh). 

Observe  that  the  base  of  the  rectangle  is  equal  to  the  base  of  the 
rhomboid,  and  the  altitude  of  the  rectangle  equal  to  the  altitude  of 
the  rhomboid. 

3.  A  rhomboid  is  equivalent  to  a  rectangle  having  the 
same  base  and  altitude.  Hence,  to  find  the  area  of  a  rhom- 
boid, find  the  area  of  a  rectangle  whose  hase  and  altitude  are 
the  same  as  the  hase  and  altitude  of  the  rhomhoid.  Or,  as  the 
rule  is  given  in  the  older  books, — "  Multiply  the  hase  hy  the 
altitude!' 


Problem. — If  the  above  figure  represents  a  piece  of  land, 
and  is  drawn  on  a  scale  of  -J-  inch  to  the  rod,  how  many 
acres  of  land  ? 

*  The  statements  upon  this  page  apply  to  the  rhombus  as  well  as  to  the 
rhomhoid. 


320  COMPLETE   ARITHMETIC. 

275.    Miscellaneous  Review. 

1.  Mr.  Watson  purchased  15  shares  of  C,  B.  &  Q.  R  R 
stock  at  12%  discount,  (a)  How  much  did  he  pay  for  the 
stock  ?  (b)  When  a  3  %  dividend  is  declared  and  paid,  how 
much  does  he  receive  ?  * 

2.  James  Cooper  bought  12  shares  of  stock  in  the  Sugar 
Grove  Creamery  at  8  %  below  par,  and  a  few  days  after  sold 
the  stock  at  5%  above  par.  How  much  more  did  he  receive 
for  the  stock  than  he  gave  for  it  ? 

3.  A  certain  city  borrowed  a  large  sum  of  money  and 
issued  therefor  10-year  5%  bonds  with  the  interest  payable 
semi-annually,  (a)  How  many  coupons  were  attached  to 
each  bond?  (b)  On  a  Si 000  bond,  each  coupon  should  call 
for  how  much  money  ? 

4.  Sometimes  such  bonds  as  those  described  in  problem  3 
are  offered  for  sale  to  the  highest  bidder,  in  "blocks"  of 
$10000,  $20000,  or  $50000.  If  a  $20000  "block"  is  "bid 
off"  at  2^%  premium,  how  much  should  the  city  receive  for 
the  block"? 

5.  What  must  be  the  nominal  value  of  5%  bonds  that 
will  yield  to  their  owner  an  annual  income  of  $750  ? 

Let  X  =  the  nominal  value ;  then  —  =  $750. 

100 

6.  What  must  be  the  nominal  value  of  4%  bonds  that 
will  yield  to  their  owner  an  annual  income  of  $720  ? 

7.  A  owns  $6000  of  5%  bonds;  B  owns  $8000  of  4|-% 
bonds.  How  much  greater  is  the  annual  income  from  B's 
bonds  than  from  A's  ? 

8.  Find  the  area  of  a  piece  of  land  in  the  form  of  a  rhom- 
boid, whose  base  is  32  rods  and  whose  altitude  is  15  rods. 

*  The  par  value  of  each  share  of  stock  mentioned  on  this  page  is  $100. 


EATIO. 

276.  Ratio  is  relation  by  quotient.  The  two  numbers 
(magnitudes)  of  which  the  ratio  is  to  be  found  are  called  the 
terms  of  the  ratio.  The  first  term  is  called  the  antecedent 
and  the  second  term  the  consequent.  The  ratio  is  the  quo- 
tient of  the  antecedent  divided  by  the  consequent. 

The  usual  sign  of  ratio  is  the  colon.  It  indicates  that  the  ratio 
of  the  two  numbers  between  which  it  stands  is  to  be  found,  the 
number  preceding  the  colon  being  the  antecedent,  and  the  number 
following  it,  the  consequent.  The  expression,  12  : 4  =  3,  is  read, 
the  ratio  of  12  to  4  is  3. 

Exercise. 
Eead  and  complete  the  following : 

1.  12:4-  4:12  =  12:2  = 

2.  18:9  --=  9:18  =  18  :  6  = 

3.  15:5  =  5:15  =  15:10  = 

Note. — It  will  be  observed  that  the  sign  of  ratio  is  the  sign  of 
division  (-=-)  with  the  line  omitted. 

277.  Every  integral  nitmher  is  a  ratio.  The  number  4  is 
the  ratio  of  a  magnitude  4  (inches,  ounces,  bushels)  to  the 
measuring  unit  1  (inch,  ounce,  bushel).  The  number  7  is 
the  ratio  of  7  yards  to  1  yard ;  of  7  dollars  to  1  dollar,  or  of 
7  seconds  to  1  second,  etc. 

Note. — The  ratio  aspect  of  numbers  is  not  the  aspect  most  fre- 
quently uppermost  in  consciousness ;  neither  ought  it  to  be.  But 
the  pupil  should  now  see  that  number  is  ratio ;  that  while  it  implies 
aggregation  and  often  stands  in  consciousness  for  magnitude,  its 
essence  is  relation — ratio. 

321 


322  COMPLETE   ARITHMETIC. 

Ratio. 

278.  Every  fractional  number  is  a  ratio.  The  fraction  |- 
is  the  ratio  of  the  magnitude  3  to  the  magnitude  4. 

So  ^,  (3),  is  the  ratio  of  12  to  4.  Observe  that  in  every  case  the 
terms  of  a  ratio  may  be  written  as  the  terms  of  a  fraction ;  the  ante- 
cedent becoming  the  numerator  and  the  consequent  the  denominator 
of  the  fraction.      The  fraction  itself  is  the  ratio. 

Exercise  L 


the  fraction  to  its  simplest  form. 

1.  The  ratio  of  20  to  6  is  \^-  =  -i/  =  3^. 

2.  The  ratio  of  6  to  20  is  ^V  =  _3_. 

3.  The  ratio  of  7  to  5  is  — ;  of  5  to  7,  — . 

4.  The  ratio  of  12  to  1  is  — ;  of  1  to  12,  - 


Exercise  II. 

1.  f  is  the  ratio  of  5  to  7 ;  of  10  to  14 ;  of  15  to  21,  etc. 

2.  ^  is  the  ratio  of  —  to  — ;  of  —  to  — ;  of  —  to  — ,  etc. 

3.  |-  is  the  ratio  of  —  to  — ;  of  —  to  — ;  of  —  to  — ,  etc. 

4.  8  is  the  ratio  of  8  to  1  ;  of  —  to  — ;  of  —  to  — ,  etc. 

Exercise  III. 
Make  the  necessary  reduction  and  find  the  ratio :  * 

1.  Of  2  feet  to  8  inches. 

2.  Of  3  yards  to  6  inches. 

3.  Of  6  rods  to  3  yards. 

4.  Of  2  rods  5  yards  to  1  yard  1  foot. 

*  The  comparison  of  two  magnitudes  involves  their  measurement  by  the  same 
standard.  To  compare  feet  with  inches,  the  inches  may  be  changed  to  feet  or  the 
feet  to  inches,  or  both  may  be  changed  to  yards. 


PART   II.  323 

Ratio. 

279.  Not  only  is  number  itself  ratio,  but  a  large  part  of 
the  work  in  arithmetic  is  merely  the  changing  of  the  form 
of  the  expression  of  ratios. 

Exercise  IV. 

(Reducing  fractions  to  their  lowest  terms.) 

1.  Express  the  ratio  of  30  to  40  in  its  simplest  form. 

2.  Express  the  ratio  of  560  to  720  in  its  simplest  form. 

3.  Express  the  ratio  of  ^    5  to  875  in  its  simplest  form. 

4.  Express  the  ratio  of  j  min.  to  2  hours  in  its  simplest 
form. 

5.  Express  the  ratio  of  1  lb.  4  oz.  to  5  lb.  8  oz.  in  its  sim- 
plest form. 

Exercise  V. 

(Reducing  improper  fractions  to  integers.) 

1.  Express  the  ratio  of  400  to  50  in  its  simplest  form. 

2.  Express  the  ratio  of  375  to  25  in  its  simplest  form. 

3.  Express  the  ratio  of  256  to  16  in  its  simplest  form. 

4.  Express  the  ratio  of  3  hours  20  minutes  to  50  minutes 
in  its  simplest  form. 

Exercise  VI. 

(Reducing  complex  fractions  to  simple  fractions.) 

1.  Express  the  ratio  of  ^  to  -|  in  its  simplest  form. 

2.  Expr(;ss  the  ratio  of  |-  to  f  in  its  simplest  form. 

3.  Express  the  ratio  of  2|-  to  8|-  in  its  simplest  form. 

4.  Express  the  ratio  of  -|-  of  an  inch  to  1  foot  in  its  sim- 
plest form. 

Note. — Observe  that  the  denominator  in  fractions  corresponds 
to  the  consequent  in  ratio. 


324  COMPLETE   ARITHMETIC. 

Ratio. 

Exercise  VII. 

(Changing  common  fractions  to  decimals.) 

1.  Express  the  ratio  of  3  to  4  (J),  in  hundredths. 

2.  Express  the  ratio  of  20  to  50,  in  tenths. 

3.  Express  the  ratio  of  30  to  80,  in  thousandths. 

4.  Express  the  ratio  of  50  sq.  rd.  to  1  acre  40  rd.,  in 
hundredths. 

Exercise  VIII. 

(Finding  what  per  cent  one  nvimber  is  of  another. ) 

1.  Express  the  ratio  of  15  to  20,  in  hundredths. 

2.  Express  the  ratio  of  14  to  200,  in  hundredths. 

3.  Express  the  ratio  of  17  to  25,  in  hundredths. 

4.  Express  the  ratio  of  16  to  33^,  in  hundredths. 

5.  Express  the  ratio  of  27  to  500,  in  hundredths. 

Exercise  IX. 

(Changing  "  per  cent "  to  a  common  fraction  in  its  lowest  terms,  or  to  a  whole 
or  mixed  number.) 

1.  A's  money  equals  40%  of  B's  money,  (a)  Express 
the  ratio  of  A's  money  to  B's  money  in  the  form  of  a  fraction 
in  its  lowest  terms,  (b)  Express  the  ratio  of  B's  money  to 
A's  money  in  its  simplest  form. 

2.  One  number  is  50%  more  than  another  number,  (a) 
Express  the  ratio  of  the;  smaller  to  the  larger  number  in  the 
form  of  a  fraction  in  its  lowest  terms,  (b)  Express  the 
ratio  of  the  larger  to  the  smaller  number  in  its  simplest 
form. 

Note. — Observe  that  the  base  in  percentage  corresponds  to  the 
consequent  in  ratio. 


PART    II.  325 

Ratio. 

Exercise  X. 

The  specific  gravity  of  a  liquid  or  solid  is  the  ratio  of  its  weight  to  the  weight  of 
the  same  bulk  of  water. 

1.  A  cubic  foot  of  water  weighs  62|-  lb.  A  cubic  foot  of 
cork  weighs  15  lb.  What  is  the  ratio  of  the  weight  of  the 
cork  to  the  weight  of  the  water  ?  Express  the  ratio  in  hun- 
dredths.    What  is  the  specific  gravity  of  cork  ? 

2.  A  certain  piece  of  limestone  weighs  37  ounces.  Water 
equal  in  bulk  to  the  piece  of  limestone  weighs  15  ounces. 
What  is  the  ratio  of  the  weight  of  the  limestone  to  the 
weight  of  the  water?  What  is  the  specific  gravity  of  the 
limestone  ? 

3.  A  certain  bottle  holds  10  ounces  of  water  or  9^  ounces 
of  oil.  What  is  the  ratio  of  the  weight  of  the  oil  to  the 
weight  of  the  water?  Express  the  ratio  in  hundredths. 
What  is  the  specific  gravity  of  the  oil? 

Note. — Observe  that  in  specific  gravity  problems,  the  weight  of 
water  corresponds  to  the  consequent  in  ratio  problems. 

280.  Miscellaneous  Questions. 

1.  What  is  the  ratio  of  a  unit  of  the  first  integral  order 
to  a  unit  of  the  first  decimal  order  ? 

2.  What  is  the  ratio  of  a  unit  of  any  order  to  a  unit  of 
the  next  lower  order  ? 

3.  What  ratio  corresponds  to  6  per  cent  ? 

4.  What  is  the  ratio  of  a  dollar  to  a  dime  ?  Of  a  dime  to 
a  cent  ?     Of  a  cent  to  a  mill  ? 

5.  What  is  the  ratio  of  1  to  y^^-?  Of  1  tenth  to  1  hun- 
dredth? 

6.  What  is  the  ratio  of  a  rod  to  a  yard?  Of  a  yard  to  a 
foot?     Of  a  foot  to  an  inch? 


326  COMPLETE    ARITHMETIC. 

Ratio. 
281.  Some  Old  Peoblems  in  New  Forms* 

1.  What  is  the  ratio  of  the  area  of  a  2 -inch  square  to  the 
area  of  a  6-in.  square  ?  *    Of  a  6-in.  square  to  a  2-in.  square  ? 

2.  What  is  the  ratio  of  the  perimeter  of  a  2 -in.  square  to 
the  perimeter  of  a  6-in.  square  ?  Of  the  perimeter  of  a  6-in. 
square  to  the  perimeter  of  a  2-in.  square  ? 

3.  What  is  the  ratio  of  the  area  of  a  3-ft.  square  to  the 
area  of  a  6-ft.  square  ?     Of  a  6-yd.  square  to  a  3-yd.  square  ? 

4.  What  is  the  ratio  of  the  perimeter  of.  a  3-ft.  square  to 
the  perimeter  of  a  6-ft.  square  ?  Of  the  perimeter  of  a  6  ft. 
square  to  the  perimeter  of  a  3-ft.  square  ? 

5.  What  is  the  ratio  of  the  solid  content  of  a  2 -inch  cube 
to  the  solid  content  of  a  6-in.  cube  ?  Of  a  6-in.  cube  to  a 
2-in.  cube  ? 

6.  What  is  the  ratio  of  the  surface  of  a  2-in.  cube  to  the 
surface  of  a  6-in.  cube  ?  Of  the  surface  of  a  6-in.  cube  to 
the  surface  of  a  3-in.  cube  ? 

7.  What  is  the  ratio  of  the  solid  content  of  a  3-ft.  cube 
to  the  solid  content  of  a  6-ft.  cube  ?  Of  a  6-yd.  cube  to  a 
3-ft.  cube  ? 

8.  What  is  the  ratio  of  the  surface  of  a  3-ft.  cube  to  the 
surface  of  a  6-ft.  cube  ?  Of  the  surface  of  a  6-yd.  cube  to 
the  surface  of  a  3-yd.  cube  ? 

9.  What  is  the  ratio  of  a  square  inch  to  a  square  foot  ?  Of 
a  cubic  inch  to  a  cubic  foot  ? 

*  If  pupils  image  the  magnitudes  compared,  they  will  find  no  difl&culty  in  the 
solution  of  these  problems. 


PART  II.  327 

Algebra. 
282.  Algebra  Applied  to  Some  Problems  in  Eatio. 
Example  I. 

The  consequent  is  c ;  the  ratio  is  r.     What  is  the  ante- 
cedent ? 

Let     X  =  the  antecedent. 

Then  -  =  r. 
c 

and     X  =  cr,  the  antecedent. 

Fro7n  the  above,  learn  that  the  antecedent  is  always  equal  to 
the  product  of  the  consequent  and  the  ratio. 

1.  Consequent  75;  ratio  11.     Antecedent? 

2.  Consequent  92;  ratio  |-.       Antecedent? 

3.  Consequent  .56  ;  ratio  ^.       Antecedent? 

Example  II. 

The  antecedent  is  a ;  the  ratio  is  r.     What  is  the  conse- 
quent ? 

Let     X  =  the  consequent. 

Then  -  =  r. 

X 

and     a  =  rx,  or  rx  =  a 
dividing  by  r,    x  =  -,  the  consequent. 

From  the  above,  learn  that  the  consequent  is  always  equal  to 
the  quotient  of  the  antecedent  divided  by  the  ratio. 

1.  Antecedent  75  ;  ratio  5.     Consequent  ? 

2.  Antecedent  9  6  ;  ratio  f.     Consequent? 

3.  Antecedent    f ;  ratio  |.     Consequent? 


328  COMPLETE    ARITHMETIC. 

283.  To  Find  Two  Numbees  when  theie  Sum  and 
Eatio  are  Given. 

Example. 

The  sum  of  two  numbers  is  36  and  their  ratio  is  3.  What 
are  the  numbers  ? 

Let  X  =  the  smaller  number. 

Then  Sx  =  the  larger  number, 

and  x-\-3x  =  3Q 

4^  =  36 

X  =  9,  the  smaller  number. 
Sx  =  27,  the  larger  number. 

Problems. 

1.  The  sum  of  two  numbers  is  196,  and  their  ratio  is  3. 
What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  294,  and  their  ratio  is  2^. 
What  are  the  numbers  ? 

3.  The  sum  of  two  decimals  is  .42,  and  their  ratio  is  2^. 
What  are  the  decimals  ? 

4.  The  sum  of  two  numbers  is  s,  and  the  ratio  of  the  larger 
to  the  smaller  is  r.     What  are  the  numbers  ? 

Let  X  =  the  smaller  number. 

Then  rx  =  the  larger  number, 

and  rx-{-  x  =  8 

or  (r  -\-l)  X  =  s 

s     ^ 


Dividing,  x 


r  +  1 


*  Observe  that  any  number  you  please  may  be  put  in  the  place  of  s,  and  any 
number  greater  than  1  in  the  place  of  r;  therefore  when  the  sum  of  two  numbers 
and  the  ratio  of  the  larger  to  the  smaller  are  given,  the  smaller  number  may  be 
found  by  dividing  the  sum  by  the  ratio  plus  1. 


PART    II, 


329 


Geometry. 
284.  The  Area  of  a  Triangle. 


1.  One  side  of  a  triangle  may  be  regarded  as  its  base. 
The  perpendicular  distance  from  its  base  (or  from  its  base 
extended)  to  the  opposite  angle  is  its  altitude. 

2.  What  is  the  altitude  of  the  first  of  the  above  triangles  ? 
Of  the  second  ?     Of  the  third  ? 

3.  Convince  yourself  by  measure- 
ment and  by  paper  cutting  that  every 
triangle  is  one  half  of  a  parallelogram 
having  the  same  base  and  the  same 
altitude  as  the  triangle. 

4.  To  find  the  area  of  a  triangle,  Find  the  area  of  the 
parallelogram  having  the  same  base  and  altitude,  and  take 
one  half  of  the  result. 


Problem. — If  the  above  figure  represents  a  piece  of  land, 
and  is  drawn  on  a  scale  of  ^  inch  to  the  rod,  what  part  of 
an  acre  of  land  does  it  represent  ? 


330  COMPLETE    ARITHMETIC. 

285.    Miscellaneous  Review. 

'  1.  The  specific  gravity  of  granite  is  2.7.*  How  much 
does  a  cubic  foot  of  granite  weigh  ? 

2.  A  certain  vessel  is  exactly  large  enough  to  contain 
1000  grains  of  water.  It  will  contain  only  700  grains  of 
petroleum.     What  is  the  specific  gravity  of  the  petroleum  ?f 

3.  The  specific  gravity  of  gold  is  19.3.  How  much  does 
a  cubic  foot  of  gold  weigh  ? 

4.  A  cubic  foot  of  sulphur  weighs  125  lbs.  What  is  the 
specific  gravity  of  sulphur  ? 

5.  A  cubic  foot  of  steel  weighs  487.5  lbs.  What  is  the 
specific  gravity  of  steel  ? 

6.  What  is  the  ratio  of  1  bu.  to  1  pk.?    Of  1  pk.  to  1  qt.? 

7.  What  is  the  ratio  of  $37^  to  $15  ?     Of  $15  to  $37|-? 

8.  What  is  the  area  of  a  rhomboid  whose  base  is  16 
inches  and  whose  altitude  is  1 6  inches  ? 

9.  Is  the  rhomboid  described  in  problem  8,  equilateral  ? 

10.  The  ratio  of  the  perimeter  of  one  square  to  the  perim- 
eter of  another  square  is  4.  What  is  the  ratio  of  the  areas 
of  the  two  squares  ? 

11.  Draw  three  triangles,  the  base  of  each  being  4  inches 
and  the  altitude  of  each  being  2  inches.  Make  one  of 
them  a  right-triangle;  another,  an  isosceles  triangle,  and  the 
third  having  angles  unhke  either  of  the  other  two.  Wliat 
can  you  say  of  the  area  of  the  right-triangle  as  compared 
with  each  of  the  others  ? 

♦  See  page  325,  exercise  10. 

t  This  means,  what  is  the  ratio  of  the  weight  of  the  petroleum  to  the  weight  of 
the  same  hulk  of  water? 


PEOPORTION. 

286.  The  terms  of  a  ratio  are  together  called  a  couplet. 
Two  couplets  whose  ratios  are  equal  are  called  a  proportion. 

The  two  couplets  of  a  proportion  are  often  written  thus:  6  :  18  = 
10 :  30,  and  should  be  read,  the  ratio  of  6  to  18  equals  the  ratio  of 
10  to  30. 

Couplets  are  sometimes  written  thus:  20  :  4  : :  50  :  10,  and  read,  20 
is  to  4  as  50  is  to  10.* 

287.   To  Find  a  Missing  Term  in  a  Proportion. 
Example  I.     36  :  12  ::.??:  25. 

The  ratio  of  the  first  couplet  is  3 ;  that  is,  the  antecedent  is  3 
times  the  consequent.  Since  the  ratios  of  the  couplets  are  equal, 
the  ratio  of  the  second  couplet  must  be  3,  and  its  antecedent  must 
be  3  times  its  consequent.     Three  times  25  =  75,  the  missing  term. 

Problems. 
Find  the  missing  term. 

1.  90:45::^:180.  4  20:60  =  ^^:225. 

2.  48:12::^:150.  5.30:50=^:175. 

3.  75  :  30  : : « :  140.  6.  90  :  20  =  « :  140.       ' 


*The  ratio  sign  (:)  may  be  regarded  as  the  sign  of  division  (-s-)  with  the  hori- 
zontal line  omitted,  and  the  proix^rtion  sign  (: :)  the  sign  of  equality  (  =  )  with  an 
erasure  through  its  center,  thus  :(=  =  ). 

331 


332  COMPLETE   ARITHMETIC. 

Proportion. 

Example  IL     36  :  12  =  48  :  x. 

Since  the  ratio  of  the  first  couplet  is  3,  the  ratio  of  the  second 
couplet  must  be  3,  and  x  must  equal  1  third  of  48.  1  third  of  48 
is  16. 

Problems. 

Find  the  missing  term. 

1.  84:21  =  172:^.  4.  20  :  60  : :  120  :a?. 

2.  96:16::    45:^.  5.  25:35=    45:^. 

3.  75:30  =  125:^.  6.  50  :  25  : :    14  :^. 

Example  III.     36:^  =  45:15. 

The  ratio  of  each  couplet  is  3  ;  so  each  consequent  must  be  1  third 
its  antecedent,  and  a:,  1  third  of  36,  or  12. 

Problems. 
Find  the  missing  term. 
1.54:^=    90:30.  4.  18  :^' : :  65  :  195. 

2.  75:^::125:25.  5.  50:^  =  12:    18. 

3.  50:a^=    40:16.  6.  35:^::  21:      3. 

Example  IV.     ^ :  12  =  100  :  25. 

The  ratio  of  each  couplet  is  4  ;  so  each  antecedent  must  be  4  times 
its  consequent,  and  a:,  4  times  12,  or  48. 

Problems. 
Find  the  missing  term. 

1.  ic:16::  51:17.  4.  ^:96  =  23:92. 

2.  ic:22  =  76:19.  5.  aj:40  ::  36  :48. 

3.  ^:11::24:    3.  6.  x:27  =  42:U. 


PART  II.  333 

288.  Practical  Problems. 

1.  If  75  yd.  of  cloth  cost  $115.25,  how  much  will  15  yd. 
cost  at  the  same  rate  ? 

75  yd.:  15  yd.  =$115.25: a-. 

2.  If  2|-  acres  of  land  cost  $76.20,  how  much  will  15  acres 
cost  at  the  same  rate  ? 

3.  If  7  tons  of  coal  can  be  bought  for  $26,  how  many  tons 
can  be  bought  for  $39  ? 

7  tons  :  x  tons  : :  <|26  :  f  39. 

4.  If  36  lb.  coffee  can  be  bought  for  $7,  how  many  pounds 
can  be  bought  for  $17 J? 

5.  If  sugar  sells  at  the  rate  of  18  lb.  for  $1,  how  much 
should  63  lb.  of  sugar  cost  ? 

6.  If  a  post  6  ft.  high  casts  a  shadow  4  feet  long,  how  high 
is  that  telegraph  pole  which  at  the  same  time  and  place  casts 
a  shadow  20  feet  long  ? 

7.  If  a  post  5  feet  high  casts  a  shadow  8  feet  long,  how 
high  is  that  steeple  which  casts  a  shadow  152  feet  long  ? 

8.  If  a  train  moves  50  miles  in  1  hr.  20  min.,at  the  same 
rate  how  far  would  it  move  in  2  hours  ? 

9.  If  a  boy  riding  a  bicycle  at  a  uniform  rate  goes  12  miles 
in  1  hr.  15  min.,  how  far  does  he  travel  in  25  minutes  ? 

To  THE  Teacher. — After  the  pupil  has  solved  the  above  prob- 
lems by  making  use  of  the  fact  of  the  equality  of  the  ratios,  he 
should  solve  them  by  an  analysis  somewhat  as  follows :  Prob.  1. 
Since  75  yd.  cost  $115.25,  1  yd.  costs  ^^  of  $115.25 ;  but  15  yd.  cost 
15  times  as  much  as  1  yd.,  so  15  yd.  cost  15  times  -^^  of  $1  ,^5.25- 


334 


COMPLETE    ARITHMETIC. 


289.    Magnitudes    Which    Are    Proportional    to    the 
Squares  of  Other  Magnitudes. 


The  areas  of  two 
squares  are  to  each 
other  as  ■  the  squares 
of  their  lengths. 


The  areas  of  two 
circles  are  to  each 
other  as  the  squares 
of  their  diameters. 


Observe  that  the  ratio  of  the  areas  of  the  above  squares  is  |  (or  |). 
But  the  area  of  each  circle  is  about  |  (more  accurately,  .785-|-)  of  its 
circumscribed  square ;  so  the  ratio  of  the  areas  of  the  circles  is  |  (or  |). 

1.  The  area  of  a  6 -inch  circle  is  how  many  times  as  great 
as  the  area  of  a  3-inch  circle  ? 

2.  If  a  4-inch  circle  of  brass  plate  weighs  3  ounces,  how 
much  will  a  6 -inch  circle  weigh,  the  thickness  being  the 
same  in  each  case  ? 

3.  If  a  piece  of  rolled  dough  1  foot  in.  diameter  is  enough 
for  17  cookies,  how  many  cookies  can  be  made  from  a  piece 
2  feet  in  diameter,  the  thickness  of  the  dough  and  the  size 
of  the  cookies  being  the  same  in  each  case  ? 

4.  If  a  piece  of  wire  ^  of  an  inch  in  diameter  will  sustain 
a  weight  of  1000  lbs.,  how  many  pounds  will  a  wire  ^  of  an 
inch  in  diameter  sustain  ? 


PART    II. 


335 


Proportion. 

290.   Magnitudes   Which   Ake    Proportional    to    the 
Cubes  of  Other  Magnitudes. 


The  solid  con- 
tents of  two  cubes 
are  to  each  other  as 
the  cubes  of  their 
lengths. 


The  solid  con- 
tents of  two  spheres 
are  to  each  other 
as  the  cubes  of 
their  diameters. 


Observe  that  the  ratio  of  the  solid  contents  of  the  above  cubes  is 
/y  (or  V).  But  the  solid  content  of  each  sphere  is  about  ^  (more 
accurately,  .5236—)  of  its  circumscribed  cube;  so  the  ratio  of  the 
solid  contents  of  the  spheres  is  g^  (or  ^). 

1.  The  solid  content  of  a  6-inch  sphere  is  how  many  times 
as  great  as  the  soHd  content  of  a  3-inch  sphere  ? 

2.  If  a  4-inch  sphere  of  brass  weighs  10  lbs.,  how  many 
pounds  will  a  6-inch  sphere  of  brass  weigh  ? 

3.  If  a  sphere  of  dough  1  foot  in  diameter  is  enough  for 
20  loaves  of  bread,  how  many  loaves  can  be  made  from  a 
sphere  of  dough  2  feet  in  diameter  ? 

4.  If  the  half  of  a  solid  8-inch  globe  weighs  4  lbs,  how 
much  will  the  half  of  a  solid  5 -inch  globe  weigh,  the  material 
being  of  the  same  quahty  ? 


336  COMPLETE    ARITHMETIC. 

291.  Magnitudes  Which  Are  Inversely  Proportional 
TO  Other  Magnitudes  or  to  the  Squares  of 
Other  Magnitudes. 

Example. 

If  5  men  do  a  piece  of  work  in  16  days,  how  long  will  it 
take  8  men  to  do  a  similar  piece  of  work  ? 

Operation  and  Explanation. 

It  is  evident  that  the  time  required  will  be  inversely  proportional 
to  the  number  of  men  employed ;  that  is,  if  twice  as  many  men  are 
employed,  not  twice  as  much,  but  |^  as  much  time  will  be  required. 
Hence  the  proportion  is  not  5  :  8  ==  16  :  x,  but,  5  :  8  =  a: :  16 ;  hence, 
5:8=10:16. 

The  interpretation  of  the  above  equation  is,  if  5  men  can  do  a 
piece  of  work  in  16  days,  8  men  can  do  it  in  10  days. 

1.  If  4  men  can  do  a  piece  of  work  in  20  days,  how  long 
will  it  take  5  men  to  do  a  similar  piece  of  work  ? 

2.  If  8  men  can  do  a  piece  of  work  in  12  days,  how  long 
will  it  take  3  men  to  do  a  similar  piece  of  work  ? 

It  can  be  shown  that  the  intensity  of  light  upon  an  object  dimin- 
ishes as  the  square  of  the  distance  between  the  luminous  body  and 
the  illuminated  object  increases ;  that  is,  if  the  distance  be  twice  as 
great  in  one  case  as  in  another,  the  intensity  is  not  twice  as  great, 
not  I  as  great,  but  i  as  great;  if  the  distances  are  as  2  to  3  the 
intensities  are,  not  as  2  to  3,  not  as  3  to  2,  but  as  9  to  4.  The 
intensity  at  2  feet  is  |  as  great  as  at  3  feet. 

3.  Object  A  is  15  feet  from  an  incandescent  electric  light. 
Object  B  is  20  feet  from  the  same  light.  Object  C  is  30 
feet  from  the  same  light,  (a)  How  does  the  intensity  of  the 
L'ght  at  B  compare  with  the  intensity  at  A  ?  (b)  How  does 
the  intensity  at  C  compare  with  the  intensity  at  A  ? 


PART    II.  337 

Algebra. 

292.    To  Find  the  Missing  Term  of  a  Proportion 
Without  Finding  the  Eatio. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms,  the  means;  thus, 
in  the  proportion  12:6  =  8:  4,  12  and  4  are  the  extremes 
and  6  and  8  are  the  means. 

Observe  that  in  the  following  proportions  the  product  of  the  means 
equals  the  product  of  the  extremes : 

6:3  =  8:4;  then  6x4  =  3x8 
^:i  =  4:2;  then  |  x  2  =  |  x  4 

Let  a  :h  =  G  :d,  stand  for  any  proportion. 

a       G  *«^ 

^^^"  ^  =  d 

Clearing  of  fractions,   ad  =  Ig 

But  a  and  d  are  the  extremes  and  h  and  g  the  means ; 
hence,  in  any  proportion  in  which  abstract  numbers  are  em- 
employed,  the  product  of  the  means  equals  the  product  of  the 
extremes. 

Example  I. 

30  :20  =  18:^. 
30a;  =  the  product  of  the  extremes. 
20  X  18  =  the  product  of  the  means. 
Then  30£C  =  20  x  18,  or  360, 

and  X  =  12. 

Example  II. 

10:25  =  ^:50. 
Then    25^  =  10  x  50,  or  500, 
and  ic  =  20. 


338  COMPLETE    ARITHMETIC. 

Algebra. 

Example  III. 

40:^  =  25:5. 
Then     25^  =  40  x  5,  or  200, 
and  ^  =  8. 

Example  IV. 

^:  35  =4:28. 
Then     2Sx  =  35  x  4,  or  140, 
and  X  —  h. 

Find  the  missing  terms : 

1.  24:72  =ic:69. 

2.  45:12  =  lb:x. 

3.  35  :  a?  =  14  :  40. 

4.  rc:70  =  3:21.  9.  .6  :  .8  =  15  :x. 

5.  55:25  =^:10.  10.  .25:5  =^:40. 

11.  If  8  acres  of  land  cost  $360,  how  much  will  15  acres 
cost  at  the  same  rate  ? 

8:15  =  360:  ;^\*' 

12.  If  12  horses  consume  3500  lb.  of  hay  per  month,  how 
many  pounds  will  1 5  horses  consume  ? 

13.  If  11  cows  cost  $280.50,  how  many  cows  can  be 
bought  for  $433.50  at  the  same  rate? 

*  Observe  that  in  the  solution  of  concrete  problems  by  the  method  here  given 
the  numbers  must  be  regarded  as  abstract.  It  would  be  absurd  to  talk  or  think  of 
finding  the  product  of  15  acres  and  360  dollars  and  dividing  this  by  8  acres.  It  is 
true,  however,  that  the  ratio  of  8  acres  to  15  acres  equals  the  ratio  of  860  dollars  to  x 
dollars.  It  is  also  true  that  in  the  proportion  8  :  15  =  360 :  x,  the  product  of  the  means 
is  equal  to  the  product  of  the  extremes. 


6. 

1 

:4  =  a;:30. 

7. 

X  ; 

:  f  =  40  :  6. 

8. 

i 

:f  =  ^:8. 

PART    II. 


339 


Geometry. 
293.  The  Area  of  a  Trapezoid. 


1.  Convince  yourself  by  measurement  and  by  paper  cut- 
ting that  from  every  trapezoid  there  may  be  cut  a  triangle 
(or  triangles)  which  when  properly  adjusted  to  another  part 
(or  parts)  of  the  trapezoid,  will  convert  the  trapezoid  into  a 
rectangle. 

2.  Convince  yourself  that  the  rectangle  made  from  a  trape- 
zoid is  not  so  long  as  the  longer  of  the  parallel  sides  of  the 
trapezoid,  and  not  so  short  as  the  shorter  of  the  parallel  sides 
of  the  trapezoid — that  its  length  is  midymy  between  the  lengths 
of  the  two  parallel  sides  of  the  trapezoid. 

Note. — Observe  that  the  length  of  the  rectangle  thus  formed  may 
be  found  by  adding  half  the  difference  of  the  parallel  sides  of  the 
trapezoid  to  its  shorter  side,  or  by  dividing  the  sum  of  its  parallel 
sides  by  2. 

3.  To  find  the  area  of  a  trapezoid,  find  the  area  of  the 
rectangle  to  which  it  is  equivalent,  or,  as  the  rule  is  usually 
given, — "Multiply  one  half  the  sum  of  the  parallel  sides  by 
the  altitude" 

4.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are 
10  inches  and  15  inches  respectively,  and  whose  altitude  is 
8  inches. 

5.  How  many  acres  in  a  trapezoidal  piece  of  land,  the 
parallel  sides  being  28  rods  and  36  rods  respectively,  and 
the  breadth  (altitude)  25  rods  ? 


340  COMPLETE    AKITHMETIC. 

294.  Miscellaneous  Review. 

1.  If  3  men  can  build  72  feet  of  sidewalk  in  a  day,  how 
many  feet  can  4  men  build  ? 

2.  If  3  men  can  do  a  piece  of  work  in  12  hours,  in  how 
many  hours  can  4  men  do  an  equal  amount  of  work  ? 

3.  If  a  piece  of  land  8  rods  square  is  worth  $500,  how  much 
is  a  piece  of  land  1 6  rods  square  worth  at  the  same  rate  ? 

4.  If  a  ball  of  yarn  3  inches  in  diameter  is  enough  for  a 
pair  of  stockings,  how  many  pairs  of  stockings  can  be  made 
from  a  ball  6  inches  in  diameter  ?  * 

5.  If  a  grindstone  12  inches  in  diameter  weighs  40  lb., 
how  much  will  a  grindstone  18  inches  in  diameter  weigh, 
the  thickness  and  quality  of  material  being  the  same  ? 

6.  The  opening  in  an  8-inch  drain  tile  is  how  many  times 
as  large  as  the  opening  in  a  2 -inch  drain  tile  ?  f 

7.  Find  the  area  of  a  rhomboidal  piece  of  land  whose 
length  (base)  is  64  rods  and  whose  width  (altitude)  is  15 
rods. 

8.  Find  the  area  of  a  trapezoidal  piece  of  land,  the  length 
of  the  parallel  sides  being  44  rods  and  52  rods  respectively, 
and  the  width  (altitude)  being  18  rods.  $ 

9.  Find  the  area  of  a  triangular  piece  of  land  whose  base 
is  42  rods  and  whose  altitude  is  20  rods. 

*  Compare  a  3-inch  cube  and  a  6-inch  cube.  Remember  that  a  3-inch  sphere  is  a 
little  more  than  half  of  a  3-inch  cube,  and  a  6-inch  sphere  a  little  more  than  one 
half  of  a  6-inch  cube. 

t  Compare  a  6-inch  square  with  a  2-inch  square.  Remember  that  a  2-inch  circle 
is  about  2  of  a  2-inch  square,  and  an  8-inch  circle  about  3  of  an  8-inch  square. 

%  Draw  a  diagram  of  the  land  on  a  scale  of  i  inch  to  the  rod. 


POWERS   AND   EOOTS. 

295.  A  product  obtained  by  using  a  number  twice  as  a 
factor  is  called  the  second  power  or  the  square  of  the  num- 
ber; thus,  25,  (5  X  5),  is  the  second  power,  or  the  square 
of  5. 

Note. — Twenty-five  is  called  the  second  power  of  5,  because  it  may 
be  obtained  by  using  5,  twice  as  a  factor.  It  is  called  the  square  of 
5,  because  it  is  the  number  of  square  units  in  a  square  whose  side  is 
5  linear  units. 

1.  What  is  the  second  power  of  2  ?     8  ?     3  ?     5  ? 

2.  What  is  the  square  of  4  ?     7?     1?     6?     9?     10? 

11'=:?  12'  =  ?  13'=?  14'  =  ? 

15'  =  ?  16'  =  ?  17'  =  ?  18'  =  ? 

(a)  Find  the  sum  of  the  eighteen  squares. 

296.  The  square  root  of  a  number  is  one  of  the  two  equal 
factors  of  the  number. 

The  radical  sign,  V,  (without  a  figure  above  it)  indicates 
that  the  square  root  of  the  number  following  it,  is  to  be 
taken  ;  thus  V  64,  means  the  square  root  of  64. 

1.  What  is  the  square  root  of  144  ?     81  ?     49  ? 

2.  What  is  the  square  root  of  36  ?     25  ?     16  ? 

V9  =  ?         ^64  =  ?        'v/121  =  ?         V100  =  ? 
V'4  =  ?        VI  =  ?  V400  =  ?        V169  =  ? 

(b)  Find  the  sum  of  the  fourteen  results. 

341 


342  COMPLETE    ARITHMETIC. 

Powers  and  Roots. 

297.  Any  number  that  can  be  resolved  into  two  equal 
factors  is  a  perfect  square. 

1.  Tell  which  of  the  following  are  perfect  squares  and 
which  are  not : 

9,  10,  12,  16,  18,  25,  32,  36. 

Note. — It  is  a  curious  fact  that  no  number,  either  integral  or 
mixed,  can  be  found  which,  when  multiplied  by  itself,  will  give  as  a 
product  10,  or  12,  or  14,  or  any  number  that  is  not  2^,  perfect  square. 

2.  Any  integral  number  that  is  a  perfect  square  is  com- 
posed of  an  even  number  of  like  prime  factors ;  that  is,  its 
prime  factors  are  an  even  number  of  2's,  3's,  5's,  7's,  etc. 

3.  Tell  which  of  the  following  are  perfect  squares  : 
144,(2  X  2x2x2x3x3);  250,(2  x5  x  5x5);  225, 

(5x5x3x3). 

Rule. — To  find  the  square  root  of  an  integral  numher^  that 
is  a  perfect  square,  resolve  the  member  into  its  prime  factors  and 
take  half  of  them  as  factors  of  the  root  ;  that  is,  one  half  as 
many  2's,  3's,  or  5's,  etc.,  as  there  are  2's,  3's,  or  5's,  etc.,  in  the 
factors  of  the  number. 

4.  Find  the  square  root  of  1225. 

1225  =  5x5x7x7.         V1225  =  5  x  7  =  35. 

5.  Find  the  square  root  of  441 ;  of  400. 

6.  Find  the  square  root  of  576  ;  of  324. 

7.  Find  the  square  root  of  784;  of  2025. 

8.  Find  the  square  root  of  625  ;  of  3025. 

(a)  Find  the  sum  of  the  last  eight  results. 


PART   II.  343 

Powers  and  Roots. 
298.  The  Square  of  Common  Fractions. 

1.  The  square  of  |,  (^  x  J),  is . 

Note. — A  square  whose  side  is  |  (of  a  Hiiear  unit)  has  an  area 
of  i  (of  a  square  unit).     Show  this  by  diagram. 

2.  Answer  the  following  and  illustrate  by  diagram  if 
necessary : 

ay  =  ?     (})'  =  ?     ay  =  ?     ay  =  ? 

(a)  Find  the  sum  of  the  eight  results. 

3.  A  square  of  sheet  brass  whose  edge  is  -^^  of  a  foot  is 
what  part  of  a  square  foot  ? 

299.  The  Square  Eoot  of  Common  Fractions. 
1.  The  square  root  of  ^  is . 

Note  1. — A  square  whose  area  is  i%  (of  a  square  unit)  is  |  (of  a 
linear  unit)  in  length.     Show  this  by  diagram. 

Note  2. — Only  those  fractions  are  perfect  squares  which,  when 
in  their  lowest  terms,  have  perfect  squares  for  numerators  and 
perfect  squares  for  denominators. 


2.  What  is  the  square  root  of  ||  ?     Of  ^  f  ?     Of  -J- 


(b)  Find  the  sum  of  the  seven  results. 

3.  The  area  of  a  square  piece  of  sheet  brass  is  -f^^  of  a 
square  foot.     What  is  the  length  of  the  side  of  the  square  ? 

4.  How  long  is  the  side  of  a  square  of  zinc  the  area  of 
which  is  -||  of  a  square  yard  ? 


344  COMPLETE    ARITHMETIC. 

Powers  and  Roots. 
300.   The  Square  of  Decimals. 

1.  The  square  of  .5  is  —  — . 

Note. — A  square  whose  side  is  .5  (of  a  linear  unit)  has  an  area  of 
.25  (of  a  square  unit).     Show  this  by  diagram. 

2.  Answer  the  following  and  illustrate  by  diagram  if  nec- 
essary : 

.1'  =  ?           .2'  =  ?           .3''  =  2  .42  =  ? 

.5*  =  ?           .6'  =  ?           .7'  =  ?  .8'  =  ? 

1.2'  =  ?         1.5'  =  ?         1.6'  =  ?  1.8'  =  ? 

(a)  Find  the  sum  of  the  twelve  results. 

3.  A  square  of  sheet  brass  whose  edge  is  .9  of  a  foot  is 
what  part  of  a  square  foot  ? 

301.   The  Square  Boot  of  Decimals. 

1.  The  square  root  of  .25  is  —  — . 

Note  1. — A  square  whose  area  is  .25  (of  a  square  unit)  is  .5  (of  a 
linear  unit)  in  length.     Show  this  by  diagram. 

Note  2. — Only  those  decimals  are  perfect  squares  which,  when  in 
their  lowest  decimal  terms,  have  numerators  that  are  perfect  squares 
and  denominators  that  are  perfect  squares.  The  decimal  denomina- 
tors that  are  perfect  squares  are  100,  10000,  1000000,  etc. 

2.  What  is  the  square  root  of  -^\%-  ?     Of  .36  ?     Of  .64  ? 
\/||4  =  ?       ^1.44  =  ?       V2.25  =  ?       ^6.25=? 

(b)  Find  the  sum  of  the  seven  results. 

3.  How  long  is  the  edge  of  a  square  of  zinc  whose  area  is 
4.84  square  feet  ?  * 

*4.84feetis}8Jfeet. 


PART   II.  345 

Powers  and  Roots. 

302.  A  product  obtained  by  using  a  number  three  times 
as  a  factor  is  called  the  third  power,  or  the  cube,  of  the 
number;  thus,  125  (5x5x5)  is  the  third  power,  or  the 
cube,  of  5. 

Note. — One  hundred  twenty-five  is  called  the  third  power  of  5, 
because  it  may  be  obtained  by  using  5  three  times  as  a  factor.  It  is 
called  the  cube  of  5  because  it  is  the  number  of  cubic  units  in  a  cube 
whose  edge  is  5  linear  units. 

V  =  1         3'  =  27        5'  =  125  V  =  343         9=*  =  729 

1.  Find  the  cube  of  12  ;  of  13 ;  of  14 ;  of  15. 
16':=?        17'  =  ?         18'  =  ?  19'=?  20'  =  ? 

(a)  Find  the  sum  of  the  nine  results. 

303.  The  cube  root  of  a  number  is  one  of  its  three  equal 
factors. 

The  radical  sign  with  a  figure  3  over  it  indicates  that  the  cube 
root  of  the  number  following  it  is  to  be  taken;  thus,  'y/512,  means, 
the  cube  root  of  512. 

Rule. — To  find  the  cuhe  root  of  an  integral  number  that  is 
a  perfect  cube,  resolve  the  member  into  its  prime  factors  and 
take  one  third  of  them  as  factors  of  the  root ;  that  is,  one  third 
as  many  2's,  3's,  or  5's,  etc.,  as  there  are  2's,  3's,  or  5's  in  the 
factors  of  the  number. 

1.  Find  the  cube  root  of  216. 

216  =  2x2x2x3x3x3.     \/216  =  2x3 

2.  Find  the  cube  root  of  1728;  of  3375;  of  2744;  of 
10648;  of  5832. 

(b)  Find  the  sum  of  the  five  results. 


346  COMPLETE    ARITHMETIC. 

Powers  and  Roots. 

304.  Miscellaneous  Problems. 

1.  Square  42.  Then  resolve  the  square  of  42  into  its 
prime  factors  and  compare  them  with  the  prime  factors  of  42. 

2.  Cube  42.  Then  resolve  the  cube  of  42  into  its  prime 
factors  and  compare  them  with  the  prime  factors  of  42. 

3.  Square  45.  Then  resolve  the  square  of  45  into  its 
prime  factors  and  compare  them  with  the  prime  factors 
of  45. 

4.  Cube  45.  Then  resolve  the  cube  of  45  into  its  prime 
factors  and  compare  them  with  the  prime  factors  of  45. 

5.  Divide  the  cube  of  15  by  the  square  of  15. 

6.  Divide  the  cube  of  ^  by  the  square  of  ^. 

7.  Divide  the  cube  of  .7  by  the  square  of  .7. 

8.  Divide  the  cube  of  2.5  by  the  square  of  2.5. 

9.  Find  the  square  root  of  5  x  5  x  7  x  7. 

10.  Find  the  cube  root  of  3x3x3x5x5x5x7x7 
X  7. 

11.  Find  the  square  root  of  each  of  the  following  perfect 
squares : 

(1)  3025         (2)  4225  (3)  5625         (4)  7225 

(^)Ui  (^)^%  iV^'A  (8)^^A 

(9)  .64  (10)  .0064       (11)  .0625      (12)  2.56  * 

12.  Find  the  cube  root  of  each  of  the  following  perfect 
cubes : 


(1)  1728 

(2)  15625 

(3)  3375 

(4)  9261 

(5)tIt 

(6)  /tV 

(7)  ,Vj 

(8)TiT 

♦  Think  of  this  number  as  fgg. 


PART    II.  347 

Algebra. 

305.  To  Find  the  Square  Eoot  of  Numbers  Eepresented 
BY  Letters  and  Figures. 

Explanation. 

Since  the  square  root  of  a  number  is  one  of  its  two  equal 
factors,  the  square  root  of  a\  (a  x  a  x  a  x  a),  is  a^,  {a  x  a). 
The  square  root  of  a^  is  a.  The  square  root  of  a^  is  a^.  Let 
a  =  3,  and  verify  each  of  the  foregoing  statements. 

1.  v"^  =  ?  V"F  =  ?  V"^  =  ?  Verify. 

2.  V  «^  =  ?  VaW  =  ?  V"^'  =  ?  Verify. 


4.  V^5^y  =  ?    •          V36a?y  =  ?  V  49^  =  ? 

5.  Let  a  =  2,  h  =  3,  and  c  =  5,  and  find  the  numerical 
value  of  each  of  the  following : 

(1)  V«^'             (2)  \/«V  (3)  VW" 


(4)  V«'^'  (5)  Va'^'c'  (6)  V(^'bV 

(a)  Find  the  sum  of  the  six  results. 

6.  Let  a  =  2,  b  =  3,x  =  5,  and  y  =  7,  and  find  the  numer- 
ical value  of  each  of  the  following : 

(1)  aV'xY     '     (2)  bV^  (3)  aV'^' 


(4)  3aVx'  (5)  4:bVy'  (6)  baV^Y 

(b)  Find  the  sum  of  the  six  results. 


348  COMPLETE   ARITHMETIC. 

Algebra. 

306.  To  Find  the  Cube  Eoot  of  Numbers  Eepresented 
BY  Letters  and  Figures. 

Explanation. 

Since  the  cube  root  of  a  number  is  one  of  its  three  equal 
factors,  the  cube  root  of  a^,  (a  x  a  x  a  x  a  x  a  X  a),  is  a\ 
(a  X  a).  The  cube  root  of  a^  is  a.  The  cube  root  of  a^  is  a^ 
Let  a  =  2,  and  verify  each  of  the  foregoing  statements. 


1.   y/V  =  ? 

i/b'  =  ^. 

\/h'  =  l 

Verify. 

2.  </«'&'  =  ? 

^a%'=:'i. 

's/a'h'  =  ? 

Verify. 

3.  V8a«  =  ? 

^27b'=l 

V64a«  =  ? 

Verify. 

4.  Let  a  =  2,  h  =  3,  and  c  =  5,  and  find  the  numerical 
values  of  each  of  the  following : 


(1)  y/a'b'  (2)  y/8c'  (3)  </27^>^^ 


(4)  2Va'¥  (5)  3V&V  (6)  4v^6V 

(a)  Find  the  sum  of  the  six  results. 

307.  Miscellaneous  Problems. 

Let  a  =  2,h  =  3,  X  =  6,  and  y  =  7,  and  find  the  numer- 
ical value  of  each  of  the  following : 


(1)  ah  +  V^Y  (2).abVxy  (3)  2a^x\/ 


(4)  ab  +  ^xY  (5)  ab^xY  (6)  2by/ xY 


(7)  \Va'y'  (8)  VK^'*  (9)  IV  aV 

(b)  Find  the  sum  of  the  nine  results. 

*  The  factors  of  this  number  are  J,  \,  a,  o,  o,  a,  &,  h. 


PART    II. 


349 


D 

E 

3 

4 

1 

2 

A                     B         C 

Qeometry. 
308.  The  Square  of  the  Sum  of  Two  Lines. 

1.  Study  the  diagram  and  observe — 

(1)  That  the  hne  AG  i^  the  sum  of 
the  lines  AB  and  BG. 

(2)  That  the  square,  1,  is  the  square 
oiAB. 

(3)  That  the  rectangle,  2,  is  as  long 
as  ^^  and  as  wide  as  BG. 

(4)  That  the  rectangle,  3,  is  as  long 
as  AB  and  as  wide  as  BG. 

(5)  That  the  square,  4,  is  the  square  of  BG. 

(6)  That  the  square,  AGED,  is  the  square  of  the  sum  of 
AB  and  BG. 

2.  SiQce  a  similar  diagram  may  be  drawn  with  any  two 
lines  as  a  base,  the  following  general  statement  may  be 
made : 

The  square  of  the  sum  of  two  lines  is  equivalent  to  the 
square  of  the  first  plus  twice  the  rectangle  of  the  two  lines 
plus  the  square  of  the  second. 

3.  If  the  hne  AB  is  10  inches  and  the  line  BG  5  inches, 
how  many  square  inches  in  each  part  of  the  diagram,  and 
how  many  in  the  sum  of  the  parts  ? 

4.  Suppose  the  hne  AB  w>  equal  to  the  hne  BG ;  what  is 
the  shape  of  2  and  3  ? 

5.  In  the  light  of  the  above  diagram  study  the  following: 

14'  =  196.  (10  +  4/  =  W+  2  (10  X  4)  +  4'  =  196. 
16'  =  ?        (10  +  6)'  = 

25'=?         (20  +  5)'  = 


350  COMPLETE    ARITHMETIC. 

309.    Miscellaneous  Review. 

1.  What  is  the  square  root  of  a^V^I 

What  is  the  square  root  of  3x3x5x5? 

2.  What  is  the  cube  root  of  dW 

What  is  the  cube  root  of  2x2x2x7x7x7? 

3.  What  is  the  square  root  of  a^V^I 
What  is  the  square  root  of  5^  x  3"? 

4.  What  is  the  cube  root  of  a%^'i 
What  is  the  cube  root  of  3'  x  5'? 

5.  The  area  of  a  certain  square  floor  is  784  square  feet. 
How  many  feet  in  the  perimeter  of  the  floor  ? 

6.  The  area  of  a  certain  square  field  is  40  acres.     How 
many  rods  of  fence  will  be  required  to  enclose  it  ? 

7.  The  solid    content  of  a  certain   cube  is   216   cubic 
inches.     How  many  square  inches  in  one  of  its  faces  ? 

8.  If  there  are  64  square  inches  in  one  face  of  a  cube, 
how  many  cubic  inches  in  its  solid  content  ? 

9.  The  square  of  (30  +  5)  is  how  many  more  than  the 
square  of  30  plus  the  square  of  5? 

10.  The  square  of  (40  +  3)  is  how  many  more  than  the 
square  of  40  plus  the  square  of  3  ? 

11.  The  square  of  a  is  a^;  the  square  of  2a  is  4al  The 
square  of  two  times  a  number  is  equal  to  how  many  times 
the  square  of  the  number  itself  ? 

12.  The  square  of  an  8-inch  line  equals  how  many  times 
the  square  of  a  4-inch  line  ? 


SQUARE   ROOT. 


310.   To  Find  the  Approximate  Square  Root  OF  Numbbrs 

THAT   ARE    NOT    PERFECT    SQUARES. 

Find  the  square  root  of  1795. 

Regard  the  number  as  representing  1795  1-inch  squares.  These 
are  to  be  arranged  in  the  form  of  a  square,  and  the  length  of  its  side 
noted.  100  1-inch  squares  =  1  10-inch  square. 

1700  1-inch  squares  =  17  10-inch  squares. 

But  16  of  the  17  10-inch  squares  can  be  arranged  in  a  square  that 
is  4  by  4;  that  is,  40  inches  by  40  inches.    See  diagram. 

After  making  this  square  (40  inches  by 
40  inches)  there  are  (1700  -  1600  -f  95)  195 
1-inch  squares  remaining.  From  these, 
additions  are  to  be  made  to  two  sides  of  the 
square  already  formed.  Each  side  is  40 
inches;  hence  the  additions  must  be  made 
upon  a  base  line  of  80  inches.  These  addi- 
tions can  be  as  many  inches  wide  as  80  is 
contained  times  in  195.*  195  ^  80  =  2+. 
The  additions  are  2  inches  wide.  These  will 
require  2  times  80,  -j-  2  times  2,  =  164  square  inches. 

After  making  this  square  (42  in.  by  42  in.)  there  are  (195  —  164)  31 
square  inches  remaining.  If  further  additions  are  to  be  made  to  the 
square,  the  31  square  inches  must  be  changed  to  tenth-inch  squares. 
In  each  1-inch  square  there  are  100  tenth-inch  squares;  in  31  square 
inches  there  are  3100  tenth-inch  squares.  From  these,  additions  are 
to  be  made  upon  two  sides  of  the  42-inch  square.  42  inches  equal 
420  tenth-inches.  The  additions  must  be  made  upon  a  base  line 
(420  X  2)  840  tenth-inches  long.  These  additions  can  be  as  many 
tenth-inches  wide  as  840  is  contained  times  in  3100.  3100  -^  840  =  3  -f . 
The  additions  are  3  tenth-inches  wide.  These  will  require  3  times 
840,  -|-  3  times  3,  =:  2529  tenth-inch  squares. 

After  making  this  square  (42.3  by  42.3)  there  are  (3100  -  2529)  571 
tenth-inch  squares  remaining.  (If  further  additions  are  to  be  made  to 
the  square,  the  571  tenth-inch  squares  must  be  changed  to  hundredth- 
inch  squares.)    The  square  root  of  1795,  true  to  tenths,  is  42.3. 

*  Allowance  must  be  made  for  filling  the  little  square  shown  at  the  upper  right- 
hand  corner  of  the  diagram. 

351 


1  1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

352 


COMPLETE    ARITHMETIC. 


Square  Root. 

IS'oTE. — Pupils  who  have  mastered  the  work  on  the  preceding 
page  will  have  no  ditficulty  in  discovering  that  the  same  result  may 
be  obtained  by  the  following  process : 

Find  the  square  root  of  1795. 
Operation. 


1795.^42.36 
16 


40  X  2 

=  80* 

•2 

195 
164 

420  x2 

=  840 
3 

3100 
2529 

4230  X  2 

=  8460 
6 

57100 
50796 

6304 


Rule. 

1.  Beginning  vjith  the 
decimal  point,  group  the  fig- 
ures as  far  as  possible  into 
periods  of  two  figures  each. 

2.  Find  the  largest  square 
in  the  left-hand  period^  and 
place  its  root  at  the  right  as 
the  first  figure  of  the  com- 
plete root. 

3.  Subtract  the  square 
from  the  left-hand  period  and  to  the  difference  annex  the  next 
period.     Regard  this  as  a  dividend. 

4.  Take  2  times  10  times  the  root  already  found  as  a  trial 
divisor,  and  find  how  many  times  it  is  contained  in  the  divi- 
dend. Write  the  quotient  as  the  second  figure  of  the  root,  and 
also  as  a  part  of  the  divisor.  Multiply  the  entire  divisor  by 
the  second  figure  of  the  root,  subtract  the  product  from  the 
dividend,  and  proceed  as  before. 

Problems. 

Find  the  approximate  square  root  (true  to  tenths) : 

(1)  875.         (2)  1526.  (3)  2754.  (4)  4150. 

(5)  624.         (6)  624.7.         (7)  62.47.         (8)  6.24. 
(a)  Find  the  sum  of  the  eight  results. 


*  The  entire  divisor  is  80  and  2;  that  is,  82. 

t  The  left-hand  period  may  consist  of  either  one  or  two  figures. 


PART   II. 


353 


Square  Root. 

311.  To  Find  the  Approximate  Squaee  Eoot  of  Decimals 

That  Are  Not  Perfect  Squares. 

Find  the  square  root  of  .6. 

Regard  the  number  as  representing  .6  of  a  1-inch  square.  .6  of  a 
1-inch  square  =  60  tenth-inch  squares. 

But  49  of  the  60  tenth-inch  squares  can  be 
arranged  in  a  square  that  is  7  by  7 ;  that  is,  7 
tenths  of  an  inch  by  7  tenths  of  an  inch. 

After  making  this  square  there  are  (60  — 
49)  11  tenth-inch  squares  remaining.  If  ad- 
ditions are  to  be  made  to  the  square,  the  11 
tenth-inch  squares  must  be  changed  to  hun- 
dredth-inch   squares.      In    each    tenth -inch 

square  there  are  100  hundredth-inch  squares;  in  11  tenth-inch 
squares  there  are  1100  hundredth-inch  squares.  From  these,  addi- 
tions are  to  be  made  upon  two  sides  of  the  .7-inch 
square.  '  .7  =  70  hundredths.  The  additions 
must  be  made  upon  a  base  line  (70  X  2)  140 
hundredth-inches  long.  These  additions  can  be 
as  many  hundredth-inches  wide  as  140  is  con- 
tained times  in  1100.  1100  -^  140  =  7+.  The 
additions  are  7  hundredth-inches  wide.  These 
will  require  7  times  140,  -f  7  times  7,  =  1029 
hundredth-inch  squares.  , 

After  making  this  square  (.77  by  .77)  there  are  (1100  -  1029)  71 
hundredth-inch  squares  remaining.  (If  further  additions  are  to  be 
made  to  the  square  the  77  hundredth-inch  squares  must  be  changed 
to  thousandth-inch  squares.)  The  square  root  of  .6,  true  to  hun- 
dredths, is  .77. 

Note. — The  work  on  this  page  should  be  first  presented  orally 
by  the  teacher.  It  must  be  given  very  slowly.  Great  care  must  be 
taken  that  pupils  image  each  magnitude  when  its  word-symbol  is 
spoken  by  the  teacher.  Any  attempt  to  move  forward  more  rapidly 
than  this  can  be  done  by  the  slowest  pupil  will  result  in  failure  so 
far  as  that  pupil  is  concerned. 


354 


COMPLETE   ARITHMETIC. 


Square  Root. 

Note. — Pupils  who  have  mastered  the  work  on  the  preceding  page 
will  readily  understand  the  following  process.     See  rule  on  page  352. 

1.  Find  the  square  root  of  .6. 


Operation. 


'6000(.774 
49 


70  X  2  =  140   1100 
7   1029 


Observe — 

1.  That  in  grouping  deci- 
mals for  the  purpose  of  ex- 
tracting the  square  root  it  is 
necessary  to  begin  at  the  deci- 
mal point. 

2.  That  the  square  root  of 
any  number  of  hundredths  is  a 
number  of  tenths;   the  square 

root  of  any  number  of  ten-thousandths  is  a  number  of  hun- 
dredths^ etc. 


11^x2  =  1540 
4 


7100 
6176 


924 


2.  Find  the  square  root  of  54264.25. 


54264.^25(232.946 
4 


20  X  2 

=  40  U 
3  11 

t2 
29 

230  X  2 

=  460  ] 
2 

L364 
924 

2320  X  2 

=  4640 
9 

440.25 
418.41 

23290  X  2 

=  4658( 
=  4658^ 

)  21.8400 
t  18.6336 

232940  X  2 

?0  3.206400 
6  2.795316 

Observe  that  the 
trial  divisor  is  al- 
ways 2  times  10 
times  the  part  of 
the  root  already 
found. 


.411084 


PART    II.  355 

Square  Root. 

312.  The  following  numbers  are  perfect  squares.  Find 
their  square  roots  by  both  the  factor  method  and  the  method 
given  on  the  four  preceding  pages. 


(1)  6889                   (2)  841 

(3)  71824 

(4)  1849                   (5)  729 

(6)  60516 

(a)  Find  the  sum  of  the  six  results. 

(7)  AV                     (8)  III 

(9)  in 

(10)  -V//            (11)  %V- 

(12)  m 

(b)  Find  the  sum  of  the  six  results. 

(13)     .81  (14)  .0625  (15)  .04 

(16)  1.21  (17)  .7921  (18)  .0004 

(c)  Find  the  sum  of  the  six  results. 

313.  Miscellaneous. 

1.  The  square  of  a  number  represented  by  one  digit  gives 
a  number  represented  by  or  digits. 

2.  The  square  of   a   number  represented  by  two  digits 
gives  a  number  represented  by  or  digits. 

3.  The  square  of  a.  number  represented  by  three  digits 
gives  a  number  represented  by  — —  or  digits. 

4.  The  square  root  of  a  perfect  square  represented  by  one 
or  two  digits  is  a  number  represented  by •  digit. 

5.  The  square  root  of   a  perfect  square  represented  by 

three  or  four  digits  is  a  number  represented  by  = 

digits. 


356  COMPLETE    ARITHMETIC. 

Square  Root. 
314.  Miscellaneous  Problems. 

1.  What  is  one  of  the  two  equal  factors  of  9216  ? 

2.  What  is  one  of  the  four  equal  factors  of  20736  ?  * 

3.  If  7921  soldiers  were  arranged  in  a  soKd  square,  how 
many  soldiers  would  there  be  on  each  side  ? 

4.  How  many  rods  of  fence  will  enclose  a  square  field 
whose  area  is  40  acres  ? 

5.  How  many  rods  long  is  one  side  of  a  square  piece  of 
land  containing  exactly  one  acre  ?  f 

6.  If  the  surface  of  a  cubical  block  is  150  square  inches, 
what  is  the  length  of  one  edge  of  the  cube  ? 

7.  How  many  rods  of  fence  will  enclose  a  square  piece 
of  land  containing  4  acres  144  square  rods  ? 

8.  Find  the  side  of  a  square  equal  in  area  to  a  rectangle 
that  is  15  ft.  by  60  ft. 

9.  Compare  the  amount  of  fence  required  to  enclose  two 
fields  each  containing  10  acres :  one  field  is  square,  and  the 
other  is  50  rods  long  and  rods  wide. 

10.  Find  the  area  of  the  largest  possible  rectangle  having 
a  perimeter  of  40  feet. 

11.  If  a  square  piece  of  land  is  ^  of  a  square  mile,  how 
much  fence  will  be  required  to  enclose  it  ? 

*  To  find  one  of  the  four  equal  factors  of  a  number  (the  4th  root)  extract  the 
square  root  of  the  square  root.    Why  ?    What  is  the  fourth  root  of  81  ? 
t  Find  the  answer  to  problem  5,  true  to  hundredths  of  a  rpd. 


PART   II.  357 

Algebra. 
315.  Square  Eoot  and  Area. 

1.  If  a  piece  of  land  containing  768  square  rods  is  three 
times  as  long  as  it  is  wide,  how  wide  is  it  ?  * 

Let  X  —  the  width, 

then  3  ^  =  the  length, 

and     X  (3  x)  or  %^  —  the  area. 

3  ^'  =  768 

x^  =  256 

X  —\^ 

2.  If  a  certain  room  is  twice  as  long  as  it  is  wide,  and  the 
area  of  the  floor  968  square  feet,  what  is  the  length  and  the 
breadth  of  the  room  ? 

3.  One  half  of  the  length  of  Mr.  Smith's  farm  is  equal  to 
its  breadth.  The  farm  contains  80  acres.  How  many  rods 
of  fence  will  be  required  to  enclose  it  ? 

4.  Each  of  four  of  the  faces  of  a  square  prism  is  an 
oblong  whose  length  is  twice  its  breadth.  The  area  of  one 
of  these  oblongs  is  72  square  inches.  What  is  the  solid 
content  of  the  prism. 

5.  The  width  of  a  certain  field  is  to  its  length  as  2  to  3. 
Its  area  is  600  square  rods.  The  perimeter  of  the  field  is 
how  many  rods  ? 

6.  If  \  of  the  length  of  an  oblong  equals  the  width  and  its 
area  is  768  square  inches,  what  is  the  length  of  the  oblong  ? 

7.  If  to  2i^  times  the  square  of  a  number  you  add  15  the 
sum  is  375.     What  is  the  number? 

*To  solve  this  problem  arithmetically,  one  must  discern  that  this  piece  of  land 
can  be  divided  into  three  equal  squares,  the  side  of  each  square  being  equal  to  the 
width  of  the  piece. 


358  COMPLETE    ARITHMETIC. 

Algebra. 
316.  Square  Eoot  and  Proportion. 

When  the  same  number  forms  the  second  and  the  third 
term  of  a  proportion  it  is  called  a  mean  proportional,  of  the 
first  and  the  fourth  term ;  thus,  in  the  proportion  3  :  6  : :  6  :  12, 
6  is  a  mean  proportional  of  3  and  12. 

Example. 

In  the  proportion  12  :  .^ : :  ^ :  75,  find  the  value  of  x. 
Since  the  product  of  the  means  equals  the  product  of  the 
extremes,  x  times  x  equals  12  times  75,  or, 

x'  =  900. 
X  =    30. 

Find  the  value  of  x  in  each  of  the  following  proportions : 

1.  9:x::x:16.  4.  12:^::^:48. 

2.  16:x::x:25.  5.     5  :^:  :;z;:  125.  * 

3.  8:x::x:S2.  6.  S6:x::x:4:9. 

(a)  Find  the  sum  of  the  six  mean  proportionals. 

7.  An  estate  was  to  be  divided  so  that  the  ratio  of  A's 
part  to  B's  would  equal  the  ratio  of  B's  part  to  C's.  If  A 
received  $8000  and  C  received  $18000,  how  much  should  B 
receive  ? 

8.  Find  the  mean  proportional  of  f  and  1|. 

9.  The  ratio  of  the  areas  of  two  squares  is  as  4  to  9. 
What  is  the  ratio  of  their  lengths  ? 

10.  The  area  of  the  face  of  one  cube  is  to  the  area  of  the 
face  of  another  cube  as  16  to  25.  What  is  the  ratio  of  the 
solid  contents  of  the  cubes  ? 


PART    II. 


359 


Geometry. 


Fig.  1. 


V\A 

W 

X> 

\ 

B 

317.  Eight-Triangles. 

1.  The  longest  side  of 
a  right-triangle  is  the 
hypothenuse.  Either  of 
the  other  sides  may  be  re- 
garded as  the  base,  and 
the  remaining  side  as  the 
perpendicular. 

2.  Convince  yourself  by 
examination  of  the  figures 

here  given,  and  by  careful  measurements 
and  paper  cutting,  that  the  square  of  the 
hypothenuse  of  a  right-triangle  is  equiva- 
lent to  the  sum  of  the  squares  of  the 
other  two  sides. 

Figures  2  and  3  are  equal  squares.  If  from 
figure  2,  the  four  right-triangles,  1,  2,  3,  4,  be 
taken,  H,  the  square  of  the  hypothenuse,  re- 
mains. If  from  figure  3,  the  four  right-tri- 
angles (equal  to  the  four  right-triangles  in 
figure  2)  be  taken,  B,  the  square  of  the  base, 
and  P,  the  square  of  the  perpendicular,  re- 
main. When  equals  are  taken  from  equals  the 
remainders  are  equal,  therefore  the  square,  H, 
equals  the  sum  of  the  squares  B  and  P. 


Fig.  2. 


Fig.  3. 


2.^^ 

P 

B 

3/ 

3.  To  find  the  hypothenuse  of  a  right-triangle  when  the 
base  and  perpendicular  are  given :  Square  the  base  ;  square 
the  perpendicular  ;  extract  the  square  root  of  the  sum  of  these 
squares. 


360  COMPLETE    ARITHMETIC. 

318.  Miscellaneous  Review. 

1.  Find  approximately  the  diagonal  of  a  square  whose 
side  is  20  feet.* 

2.  Find  approximately  the  distance  diagonally  across  a 
rectangular  floor,  the  length  of  the  floor  being  30  feet  and 
its  breadth  20  feet. 

3.  How  long  a  ladder  is  required  to  reach  to  a  window  25 
feet  high  if  the  foot  of  the  ladder  is  6  feet  from  the  building 
and  the  ground  about  the  building  level  ? 

4.  If  the  length  of  a  rectangle  is  a,  and  its  breadth  h, 
what  is  the  diagonal  ? 

5.  The  base  of  a  right  triangle  is  40  rods  and  its  perpen- 
dicular, 60  rods,  (a)  What  is  its  hypothenuse  ?  (b)  What 
is  its  area  ?     (c)  What  is  its  perimeter  ? 

6.  The  area  of  a  certain  square  piece  of  land  is  2^  acres, 
(a)  Find  (in  ¥ods)  its  side,  (b)  Find  its  perimeter,  (c) 
Find  its  diagonal,  true  to  tenths  of  a  rod. 

7.  The  length  of  a  rectangular  piece  of  land  is  to  its 
breadth  as  4  to  3.  Its  area  is  30  acres,  (a)  Find  its 
breadth,  (b)  Find  its  perimeter,  (c)  Find  the  distance 
diagonally  across  it. 

8.  A  certain  piece  of  land  is  in  the  shape  of  a  right-tri- 
angle. Its  base  is  to  its  altitude  as  3  to  4.  Its  area  is  96 
square  rods,  (a)  Find  the  base,  (b)  Find  the  altitude, 
(c)  Find  the  perimeter. 

9.  Find  one  of  the  two  equal  factors  of  93025. 

*From  the  study  of  right-triangles  on  page  359  it  may  be  learned  that  the 
diagonal  of  a  square  is  equal  to  the  square  root  of  twice  the  square  of  its  side. 


METEIC   SYSTEM. 

Note. — The  teacher  should  present  this  subject  orally  before 
attempting  the  pages  that  follow.  It  will  be  well  if  this  oral  work 
can  be  commenced  many  weeks  before  this  page  is  reached  in  the 
regular  work  of  the  class.  A  meter  stick,  a  liter  measure,  and  metric 
weights  should  be  provided  and  each  pupil  should  weigh  and  meas- 
ure until  he  can  easily  think  quantity  in  the  units  of  this  system 
without  reference  to  the  units  of  any  other  system. 

319.  All  units  in  the  metric  system  of  measures  and 
weights  are  derived  from  the  primary  unit  known  as  the 
meter. 

When  the  length  of  the  primary  unit  of  this  system  was  deter- 
mined it  was  supposed  to  be  one  ten-millionth  of  the  distance  from 
the  equator  to  the  pole.  A  pendulum  that  vibrates  seconds  is  nearly 
one  meter  long. 

In  the  names  of  the  derived  units  of  this  system  the  prefix  deka 
means  10;  hekto  means  100;  kilo  means  1000;  myria  means  10000; 
deci  means  tenth ;  centi  means  hundredth ;  milli  means  thousandth. 

320.   Linear  Measure. 

10  millimeters  (mm.)  =  1  centimeter  (cm.).* 

10  centimeters  =  1  decimeter  (dm.). 

10  decimeters  =  1  meter  (m.). 

10  meters  —  1  dekameter  (Dm.). 

10  dekameters  =  1  hektometer  (Hm.). 

10  hektometers  =  1  kilometer  (Km.). 

10  kilometers  =  1  myriameter  (Mm.). 

*  In  the  common  pronunciation  of  tliese  words  the  primary  accent  is  on  the  first 
syllable  and  a  secondary  accent  on  the  penultimate  syllable ;  thus,  cen'tim6ter.  In 
the  better  pronunciation  the  accent  is  on  the  vowel  preceding  the  letter  m,  that  is, 
on  the  antepenultimate  syllable ;  thus,  centim'eter,  dekam'eter,  etc. 


362  .         COMPLETE    ARITHMETIC. 

Metric  System. 

321.  The  names  of  the  units  of  surface  measurement  are 

the  same  as  those  used  for  hnear  measurement,  combined 
with  the  word  square;  thus,  a  surface  equivalent  to  a  square 
whose  side  is  a  meter  is  1  square  meter. 

The  pupil,  if  properly  taught  to  this  point,  will  be  able,  without 
difficulty,  to  fill  the  blanks  iu  the  table  of — 

Squaee  Measure. 

100  square  millimeters  (sq.  mm.)  =  1  square  centimeter  (sq.  cm.). 

square  centimeters  =  1  square  decimeter  (sq  dm.). 

square  decimeters  =  1  square  meter  (sq.  m.). 

square  meters  =  1  square  dekameter  (sq.  Dm.). 

square  dekameters  =  1  square  hektometer  (sq.  Hm.). 

square  hektometers  ,           =  1  square  kilometer  (sq.  Km.). 

square  kilometers  =  1  square  myriameter  (sq.  Mm.). 

Note. — The  special  unit  of  surface  measure  for  measuring  land 
is  equivalent  to  a  square  whose  side  is  ten  meters.  Thigi  unit  is 
called  an  ar. 

100  centars  (ca.)  =  1  ar  (a.). 

100  ars  =1  hektar  (Ha.). 

Exercise. 

1.  In  a  square  decimeter  there  are  sq.  cm. 

2.  In  2  square  decimeters  there  are  sq.  cm. 

3.  In  a  2 -decimeter  square  there  are  sq.  cm. 

4.  In  a  square  meter  there  are  sq.  dm. 

5.  In  2  square  meters  there  are  sq.  dm. 

6.  In  a  2 -meter  square  there  are  sq.  dm. 

7.  In  a  square  meter  there  are  sq.  cm. 

8.  In  2  square  meters  there  are  sq.  cm. 

9.  In  a  2 -meter  square  there  are  sq.  cm. 


PART    II.  363 

Metric  System. 

322.  The  names  of  the  units  of  volume  measurement  are 

the  same  as  those  used  for  linear  measurement,  combined 
with  the  word  cuMc;  thus,  a  volume  equivalent  to  a  cube 
whose  edge  is  a  meter  is  1  cubic  meter. 

The  pupil  should  be  able  easily  to  fill  the  blanks  in  the  table  of — 

Volume  Measure. 

1000  cubic  milhmeters  (cu.  mm.)  =  1  cubic  centimeter  (cu.  cm.).* 

cubic  centimeters  =  1  cubic  decimeter  (cu.  dm.). 

——  cubic  decimeters  =  1  cubic  meter  (cu.  m.). 

cubic  meters  =  1  cubic  dekameter  (cu.  Dm.). 

cubic  dekameters  =  1  cubic  hektometer  (cu.  Hm.). 

Note  1. — The  special  unit  of  capacity  for  measuring  liquids, 
grain,  small  fruits,  etc.,  is  the  liter.  It  is  equal  to  1  cubic  decimeter. 
10  liters  (1.)  =  1  dekaliter  (Dl.),  and  1  tenth  of  a  liter  =  1  deciliter 
(dl.),  etc. 

Note  2. — The  special  unit  for  measuring  wood  is  the  ster.  It  is 
equal  to  1  cubic  meter. 

Exercise. 

1.  A  cubic  meter  equals  liters. 

2.  A  cubic  meter  equals  cubic  decimeters. 

3.  A  cubic  meter  equals  cubic  centimeters. 

4.  A  cubic  decimeter  equals  cubic  centimeters. 

5.  Two  cubic  decimeters  equal  cubic  centimeters. 

6.  A  2-decimeter  cube  equals  cubic  centimeters. 

7.  A  5-centimeter  square  equals square  centimeters. 

8.  A  5 -centimeter  cube  equals  cubic  centimeters. 

9.  One  tenth  of  a  liter  equals  cubic  centimeters. 

10.  One  deciliter  equals  cubic  centimeters. 

*  The  abbreviation  cc.  is  often  used  for  cubic  centimeter. 


364  COMPLETE   ARITHMETIC. 

Metric  System. 

323.  The  primary  unit  of  weight  is  the  gram.  This  equals 
the  weight  of  one  cubic  centimeter  of  pure  water. 

Weight. 

10  miUigrams  (mg.)  =  1  centigram  (eg.). 
10  centigrams  =  1  decigram  (dg.). 

10  decigrams  =  1  gram  (g.). 

10  grams  =  1  dekagram  (Dg.). 

10  dekagrams  =  1  hektogram  (Hg.). 

10  hektograms  =  1  kilogram  (Kg.). 

Note. — The  special  miit  for  the  weight  of  very  heavy  articles  is 
the  tonneau.  It  equals  the  weight  of  a  cubic  meter  of  pm'e  water, 
or  1000  kilograms. 

Exercise. 

1.  The  weight  of  1  liter  of  water  is  grams. 

2.  The  weight  of  6  cubic  centimeters  of  water  is  . 

3.  The  weight  of  a  cubic  decimeter  of  water  is  . 

4.  One  kilogram  of  water  equals  cubic  centimeters. 

5.  One  hektogram  of  water  equals  cc. 

6.  One  dekagram  of  water  equals cubic  centimeters. 

7.  If  the  specific  gravity   of  iron  is   7.5,  what  is  the 
weight  of  a  cubic  centimeter  of  iron  ? 

8.  If  the  specific  gravity  of  cork  is  ^,  what  is  the  weight 
of  a  cubic  decimeter  of  cork  ? 

9.  If  the  specific  gravity  of  oil  is  .9,  what  is  the  weight 
of  a  liter  of  oil  ? 

10.  What  is  the  weight  of  a  cubic  meter  of  stone  whose 
specific  gravity  is  2.5  ? 

11.  What  is  the  weight  of  a  cubic  decimeter  of  wood 
whose  specific  gravity  is  .8  ? 


PART  II.  365 

Metric  System.' 
324.  Miscellaneous  Problems. 

1.  Find  the  area  of  a  rectangular  surface  that  is  1  meter 
long  and  6  decimeters  wide.  Make  a  diagram  of  this  surface 
upon  the  blackboard. 

2.  Find  the  area  of  a  rectangular  surface  that  is  2  deci- 
meters long  and  5  centimeters  wide.  Make  a  diagram  of 
this  surface  on  your  slate  or  paper. 

3.  Find  the  solid  content  of  a  5-centimeter  cube.  A  5- 
centimeter  cube  is  what  part  of  a  cubic  decimeter  ? 

4.  Find  the  solid  content  of  a  4-decimeter  cube.  A  4- 
decimeter  cube  is  what  part  of  a  cubic  meter  ? 

5.  Find  the  entire  surface  of  a  4-centimeter  cube.  The 
surface  of  a  4  centimeter  cube  is  what  part  of  a  square  deci- 
meter ? 

6.  Find  the  area  of  a  rectangular  surface  that  is  2.4  yards 
by  5  yards ;  of  a  rectangular  surface  that  is  2.4  meters  by  5 
meters. 

7.  Which  is  the  larger  of  the  two  surfaces  described  in 
problem  6  ? 

8.  Find  the  area  of  a  rectangular  surface  that  is  3.5  yards 
by  2.5  yards ;  of  a  rectangular  surface  that  is  3.5  meters  by 
2.5  meters. 

9.  Find  the  volume  of  a  rectangular  solid  that  is  3.4  feet 
by  3  feet  by  2  feet ;  of  a  rectangular  solid  that  is  3.4  meters 
by  3  meters  by  2  meters. 

10.  Find  the  volume  of  a  rectangular  solid  that  is  3.5 
meters  by  2.3  meters  by  4.6  meters. 

11.  What  is  the  weight  of  a  cubic  decimeter  of  wood 
whose  specific  gravity  is  .5  ? 


366  COMPLETE    ARITHMETIC. 

•  Metric  System. 
325.  Miscellaneous  Problems. 

1.  Estimate  in  meters  the  width  of  the  lot  upon  which 
the  school  building  stands.     Measure  it. 

2.  Estimate  in  centimeters    the    width   of   your   desk. 
Measure  it. 

3.  Estimate  in  square  centimeters  the  area  of  a  sheet  of 
paper.     Measure  and  compute. 

4.  Estimate  in  square  meters  the  area  of  the  blackboard. 
Measure  and  compute. 

5.  Estimate  the  number  of  cubic  meters  of  air  in  the 
school  room.     Measure  and  compute. 

6.  Estimate  in  grams  the  weight  of  a  teaspoonful  of 
water.     Weigh  it.* 

7.  Estimate  in  kilograms  your  own  weight. 

8.  Estimate  in  liters  the  capacity  of  a  water  pail. 

9.  Estimate  in  kilograms  the  weight  of  a  gallon  of  water. 
10.  Estimate  in  kilometers  the  distance  from  the  school- 
house  to  your  home. 

326.  Table  of  Equivalents. 

Meter a  little  more  than  1  yard  .    .    .  39.37  inches. 

Kilometer    ....  nearly  |  of  a  mile 3280.8-f-  feet. 

Decimeter   ....  nearly  4  inches 3.937  inches. 

Ar nearly  ^  of  an  acre  .....  3.954  sq.  rd. 

Ster a  little  more  than  I  cord  .    .    .  35.3-|-  cu.  ft. 

Liter a  little  more  than  1  liquid  quart,  1.056 -f  qt. 

Gram nearly  15^  grains 15.4+ grains. 

Kilogram nearly  2  J  pounds 2.204-|- lb. 

*Every  school  should  be  provided  with  scales,  weights,  and  measures. 


PART    II.  367 

Algebra. 
327.  Metric  Units  in  Algebraic  Problems. 

1.  I  am  thinking  of  a  rectangular  surface.  Its  length  is 
5  times  its  breadth.  Its  area  is  45  square  decimeters.  How 
long  and  how  wide  is  the  surface  ?  * 

2.  I  am  thinking  of  a  triangular  surface.  Its  base  is 
three  times  its  altitude.  Its  area  is  8.64  square  meters. 
What  is  the  length  of  its  base  ? 

3.  I  am  thinking  of  a  cube  whose  entire  surface  is  150 
square  centimeters.     What  is  the  length  of  one  of  its  edges  ? 

4.  The  perimeter  of  a  certain  rectangle  is  20.4  meters. 
Its  length  is  twice  its  breadth,  (a)  Find  its  length  and 
breadth,     (b)  Find  its  area. 

5.  The  difference  in  the  weight  of  two  lead  balls  is  24 
grams.  The  united  weight  of  the  two  balls  is  1  kilogram, 
(a)  Find  the  weight  of  each  ball,  (b)  Does  the  heavier 
ball  weigh  more  or  less  than  1  pound  ? 

6.  A  merchant  had  three  pieces  of  lace.  In  the  second 
piece  there  were  twice  as  many  meters  as  in  the  first.  In 
the  third  piece  there  were  6  meters  more  than  in  the  second. 
In  the  three  pieces  there  were  106  meters,  (a)  How  many 
meters  in  each  piece  ?  (b)  Were  there  more  or  less  than  53 
yards  in  the  second  piece  ? 

7.  John  weighs  3.6  kilograms  more  than  Henry.  To- 
gether they  weigh  83.6  kilograms,  (a)  Find  the  weight  of 
each  boy.  (b)  Does  John  weigh  more  or  less  than  90 
pounds  ? 

*  Let  X  =  the  number  of  decimeters  in  the  breadth  of  the  surface. 


368  COMPLETE    ARITHMETIC. 

Algebra. 
328.  Metric  Units  in  Algebraic  Problems. 

1.  A  ball  rolling  down  a  perfectly  smooth  and  uniformly 
inclined  plane  rolls  3  times  as  far  the  2nd  second  as  the  1st; 
5  times  as  far  the  3rd  second  as  the  1st;  7  times  as  far  the 
4th  second  as  the  first.  If  in  4  seconds  it  rolls  192  deci- 
meters (a)  how  far  did  it  roll  in  the  1st  second  ?  (b)  In  the 
4th  second  ?  (c)  Did  it  roll  more  or  less  than  48  inches  in 
the  first  second  ? 

2.  I  am  thinking  of  a  right-triangle.  Its  altitude  is  to  its 
base  as  3  to  4.  The  sum  of  its  altitude  and  base  is  14  centi- 
meters, (a)  Find  the  altitude,  (b)  Find  the  base,  (c) 
Find  the  area,  (d)  Find  the  hypothenuse.  (e)  Is  the 
hypothenuse  more  or  less  than  4  inches  ? 

3.  A  freely  falling*  body  falls  three  times  as  far  the  2nd 
second  of  its  fall  as  it  does  the  1st  second.  In  two  seconds 
it  falls  19.6  meters,  (a)  How  far  does  it  fall  in  the  1st 
second?     (b)  In  the  2nd  second? 

4.  A  freely  falling  body  falls  3  times  as  far  the  2nd  minute 
of  its  fall  as  it  does  the  1st  minute.  In  two  minutes  it  falls 
70560  meters,  (a)  How  far  does  it  fall  in  the  1st  minute  ? 
(b)  In  the  2nd  minute?  (c)  70560  meters  equals  how  many 
kilometers?  (d)  70560  meters  equals  (approximately)  ho\i» 
many  miles? 

5.  A  freely  falling  body  falls  3  times  as  far  the  2nd  half- 
second  as  it  does  the  1st  half-second.  In  one  second  it  falls 
4.9  meters,  (a)  How  far  does  it  fall  in  the  1st  half-second  ? 
(b)  In  the  2nd  half -second  ? 

*  A  freely  falling  body  is  a  body  falling  in  a  perfect  vacuum. 


PART  II.  369 

Geometry. 
329.   The  Circumference  of  a  Circle. 

1.  Cut  a  3-iiich  circle  from  cardboard.  By  rolling  it  upon 
a  foot  rule,  measure  its  circumference. 

2.  Measure  the  diameter  of  a  bicycle  wheel;  then  by 
rolHng  it  upon  the  ground  or  upon  the  school-room  floor, 
measure  its  circumference. 

3.  In  a  similar  manner  measure  the  diameters  and  the 
circumferences  of  other  wheels  until  you  are  convinced  that 

the  circumference  of  a  circle  is  a  little  more  than times 

its  diameter. 

4.  The  circumference  of  a  circle  is  nearly  3^  times  the 
diameter;  more  accurately,  it  is  3.141592+  times  the 
diameter. 

I^OTE. — It  is  a  curious  fact  that  the  diameter  of  a  circle  being 
given  in  numbers,  it  is  impossible  to  express  in  numbers  its  exact 
circumference.  The  circumference  being  given  in  numbers,  it  is 
impossible  to  express  in  numbers  its  exact  diameter.  In  other 
words,  the  exact  ratio  of  the  circumference  to  the  diameter  is  not 
expressible. 

5.  Find  the  approximate  circumference  of  a  5-inch  circle ; 
of  a  7-inch  circle;  of  a  10-inch  circle.* 

6.  Find  the  approximate  diameter  of  a  circle  that  is  6  ft. 
in  circumference.* 

7.  The  circumference  of  a  6-inch  circle  is  how  many  times 
the  circumference  of  a  3-inch  circle  ? 

8.  The  diameter  of  a  circle  whose  circumference  is  12 
inches  is  what  part  of  the  diameter  of  a  circle  whose  circum- 
ference is  24  inches  ? 

*  In  the  solution  of  such  problems  as  these,  the  pupil  may  use,  as  the  approxi- 
mate ratio  of  the  circumference  to  the  diameter,  3.14. 


CONTENTS— PART  III, 


Denominate  Numbers, 

Linear  Measure, 

Surface  Measure, 

Volume  Measure, 

Capacity,   -  -  -  - 

Weight,  _  .  .  . 

Time,  .  .  .  .. 

Circular  Measure,    - 

Longitude  and  Time, 

Value  Measure, 
Short  Methods, 

Multiplication, 

Division,    -  -  -  - 

Cancellation, 

Miscellaneous,     - 
Practical  Approximations, 
Miscellaneous  Examination  Problems, 
Explanatory  Notes, 


Pages 

371-400 

371,372 

373-380 

381-390 

391,  392 

393-395 

396,397 

398 

399 

400 

401-414 

401-409 

410 

411, 412 

412-414 

415-420 

421-442 

443-446 


370 


PART   III. 

DENOMINATE   NUMBEKS. 

Linear  Measure. 

Note. — Pupils  who  have  mastered  the  Elementary  Book  and  the 
preceding  pages  of  this  book  have  had  much  practice  in  the  use  of 
denominate  numbers.  In  part  to  provide  for  ready  reference  and  in 
part  to  give  further  application  of  the  principles  already  presented, 
the  subject  is  here  treated  as  a  whole. 

331.  The  English  and  United  States  standard  unit  of 
length  is  the  Imperial  yard  arbitrarily  fixed  by  Act  of 
Parliament  and  afterward  adopted  in  the  United  States.  It 
is  about  14x1^1"  ^^  ^^®  length  of  a  pendulum  that  vibrates 
once  a  second  at  the  level  of  the  sea  in  the  latitude  of 
London.     It  is  ff f f  of  a  meter. 

Table. 

12  inches  (in.)  =  1  foot  (ft.). 

3  feet  =  1  yard  (yd.). 

5 1  yards  =  1  rod  (rd.). 

16^  feet  =  1  rod. 

320  rods  =  1  mile  (mi.). 

1760  yards         =  1  mile. 

5280  feet  =  1  mile. 

1  fathom  (used  in  measuring  the  depth  of  the  sea)  =  6  feet. 
1  knot  (used  in  navigation)  =  1.15+  miles. 

1  league  (used  in  navigation)  =  3  knots. 

1  hand  (used  in  measmdng  the  heights  of  horses)       =  4  inches. 
1  chain  (used  by  civil  engineers)  =  100  feet. 

1  chain  (used  by  land  surveyors)  =  66  feet. 

1  pace  (used  in  measuring  approximately)  =  ^  of  a  rod. 

1  barleycorn  (used  in  grading  length  of  shoes)  =  J  of  an  inch. 

1  furlong  (a  term  nearly  obsolete)  =  ^  of  a  mile. 

371 


372  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Linear  Measure. 

Exercise. 

1.  Mont  Blanc  is  15810  feet,  or  about  miles  high. 

2.  Mt.  Everest  is  29000  feet,  or  about  miles  high. 

3.  Commodore  Dewey  opened  fire  on  the  enemy  at  a 
distance  of  5000  yards,  or  about  miles. 

4.  My  horse,  measured  over  the  front  feet,  is  16^  hands, 
or  feet  inches  high. 

5.  The  vessel  seemed  to  be  about  three  leagues,  or  

miles  distant. 

6.  On  sounding,  they  found  the  depth  of  the  water  to  be 
15  fathoms,  or  feet. 

7.  The  cruiser  made  20  knots,  or  about  miles,  an 

hour. 

8.  The  length  of  the  lot  was  36  paces,  or  about rods. 

9.  10000  feet  is  nearly  miles. 

10.  15000  feet  is  nearly  miles. 

11.  1000  yards  is  about  of  a  mile. 

12.  100  feet  is  rods  foot. 

13.  200  feet  is  - —  rods  feet. 

14.  300  feet  is  rods  feet. 

15.  A  kilometer  is  about  rods. 

16.  A  Civil  Engineer's  chain  is  rods  foot. 

Problems. 

1.  A  seven-foot  drive  wheel  of  a  locomotive  makes  how 
many  revolutions  to  the  mile  ? 

2.  Which  is  the  longest  distance,  5  miles  319  rods  16 
feet  6  inches,  5  miles  319  rods  5  yards  1  foot  6  inches,  or  6 
miles  ? 

3.  Eeduce  40  rd.  4  ft.  5  in.  to  inches. 


PART    III.  373 

Denominate  Numbers — Surface  Measure. 
332.  The  standard  unit  of  surface  measure  is  a  square 
yard  which  is  the  equivalent  of  a  1-yard  square.  This 
unit,  like  the  square  foot,  square  inch,  square  rod,  and 
square  mile,  is  derived  from  the  corresponding  unit  of  linear 
measure. 

Table. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30i  square  yards  =  1  square  rod  (sq.  rd.). 

272 1  square  feet  =  1  square  rod. 

160  square  rods  =  1  acre  (A.). 

4840  square  yards  =  1  acre. 

43560  square  feet  =  1  acre. 

640  acres  =  1  square  mile  (sq.  mi.). 

Exercise. 

1.  Show  by  a  drawing  that  there  are  144  square  inches  in 
a  1-foot  square. 

2.  Show  by  a  drawing  that  there  are  9  square  feet  in  a 
1-yard  square. 

3.  Show  by  a  drawing  that  there  are  30^  square  yards  in 
a  1-rod  square. 

4.  Estimate  the  number  of  square  yards  of  blackboard  in 
the  room ;  the  number  of  square  feet  of  blackboard. 

5.  Estimate  the  number  of  square  feet  in  the  floor  of  the 
schoolroom  ;  the  number  of  square  yards. 

6.  Estimate  the  square  yards  of  plastering  on  the  walls  of 
the  schoolroom. 

7.  Estimate  the  number  of  square  rods  in  the  schoolhouse 
lot.     Is  the  lot  more  or  less  than  ^  of  an  acre  ? 


374  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Surface  Measure. 

333.  In  the  measurement  of  land  it  is  more  convenient 
to  use  a  decimal  scale ;  hence  the  invention  of  the  Gunter 
Chain.  This  chain  is  4  rods  long  and  is  divided  into  100 
links. 

Observe  that  links  are  hundredths  of  chains. 
Observe  that  square  chains  are  tenths  of  acres. 

1.  Land,  3  chains  by  4  chains  contains  acres. 

2.  Land,  5  chains  by  4  chains  contains  acres. 

3.  Land,  3  chains  by  8  chains  contains  acres. 

4.  Land,  5  chains  by  7  chains  contains  acres. 

5.  Land,  8  chains  by  6  chains  contains  acres. 

6.  Two  chains  35  links  equals  chains. 

7.  Two  chains  75  links  equals  chains. 

8.  Two  chains  5  links  equals  chains. 

9.  Two  chains  9  links  equals  chains. 

10.  Land,  4  ch.  by  4.50  ch.  contains  acres. 

11.  Land,  5  ch.  by  3.20  ch.  contains  acres. 

12.  Make  a  rule  and  find  the  number  of  acres  in  each  of 
the  following : 

(1)  Land,  12  chains  35  links  by  9  chains  50  links. 

(2)  Land,  21  chains  8  links  by  12  chains  30  links. 

(3)  Land,  32  chains  25  links  by  15  chains  6  links. 

(a)  Find  the  sum  of  the  area  of  the  ten  pieces  of  land 
described  on  this"^age. 

To  THE  Teacher. — A  rod  is  exactly  25  hnks.  A  foot  is  about 
\\  links.  Hence  rods  and  feet  can  be  easily  changed  to  chains  and 
links  by  regarding  each  4  rods  as  1  chain  and  each  additional  rod  as 
25  links  and  each  additional  foot  as  1^  links.  The  error  in  any  one 
measurement  never  exceeds  2  inches.  9  rd.  12  ft.  =  2  chains  43 
(25  + 18)  links. 


PART   III.  375 

Denominate  Numbers— Surface  Measure. 

334.  To  determine  the  amount  of  carpet  necessary  for  a 
given  room  several  minor  problems  must  be  solved  which 
can  be  best  studied  by  means  of  an — 

Example. 

1.  How  many  yards  of  carpet  must  be  purchased  for  a 
room  1 6  ft.  by  2  0  ft.  if  the  carpet  is  1  yd.  wide  ? 

(1)  How  many  breadths  will  be  necessary  if  the  carpet 
is  put  down  lengthwise  of  the  room  ?  How  much  must  be 
cut  off  or  turned  under  from  one  breadth  in  this  case  ? 

(2)  How  many  breadths  will  be  necessary  if  the  carpet 
is  put  down  crosswise  of  the  room?  How  much  must  be 
cut  off  or  turned  under  from  one  breadth  in  this  case  ? 

(3)  Make  two  diagrams  of  the  room  on  a  scale  of  1  inch 
to  the  foot  and  show  the  breadths  of  carpet  in  each  case. 

(4)  How  many  yards  must  be  purchased  in  each  case  ? 

(5)  If  in  the  first  case  there  is  no  waste  in  matching  the 
figure  and  in  the  second  case  there  is  a  waste  of  8  inches 
on  each  breadth  exce;pt  the  first,  which  plan  of  putting  down 
the  carpet  will  require  the  greater  number  of  yards  ? 

(6)  If  the  carpet  costs  90^  a  yard  and  the  conditions  are 
as  stated  in  No.  5,  what  is  the  cost  of  the  carpet  in  each 
case  ? 

2.  How  many  yards  of  carpet  must  be  purchased  for  a 
room  16  ft.  by  20  ft.  if  the  carpet  is  f  of  a  yard  wide  and 
there  is  no  waste  in  matching  the  figure  ? 

3.  How  many  yards  of  carpet  must  be  purchased  for  a 
room  that  is  15*  ft.  6  in.  by  16  ft.  4  in.  if  the  carpet  is  f  of  a 
yard  wide,  is  put  down  lengthwise  of  the  room,  and  there  is 
no  waste  in  matching  the  figure  ? 


376  COMPLETE   ARITHMETIC. 

Denominate  Numbers. 
335.     Plastering  and  Papering. 

1.  How  many  square  yards  of  plastering  in  a  room  (walls 
and  ceiling)  that  is  15  ft.  by  18  ft.  and  12  ft.  high,  an  allow- 
ance of  1 2  square  yards  being  made  for  openings  ? 

Note. — In  estimating  the  cost  of  plastering,  allowance  is  made 
for  "  openings  "  (windows  and  doors)  only  when  they  are  very  large 
in  proportion  to  the  wall  to  be  covered.  Why  are  plasterers 
unwilling  to  deduct  the  entire  area  of  all  the  openings  ? 

2.  At  24^  a  square  yard  how  much  will  it  cost  to  plaster 
a  room  that  is  17  ft.  by  20  ft.  and  10  feet  from  the  floor  to 
the  ceiling,  deducting  1 6  square  yards  for  openings  ? 

3.  How  many  "double  rolls"  of  paper  will  be  required 
for  the  walls  of  a  room  that  is  14  ft.  by  16  ft.  and  11  ft. 
high  above  the  baseboards,  if  an  allowance  of  1  full  "double 
roll"  is  made  for  openings  ? 

Note. — Wall  paper  is  usually  18  inches  wide.  A  "single  roll" 
is  24  ft.  long.  A  "  double  roll "  is  48  ft.  long.  In  papering  a  room 
11  ft.  high  it  would  be  safe  to  count  on  4  full  strips  from  each 
"  double  roll."  The  remnant  would  be  valueless  unless  it  could  be 
used  over  windows  or  doors.  Since  each  strip  is  18  inches  wide,  a 
"  double  roll "  will  cover  72  inches  (6  ft.)  of  wall  measured  hori- 
zontally. 

4.  At  12^  a  "single  roll,"  how  much  will  the  paper  cost 
for  the  walls  of  a  room  that  is  12  ft.  by  14  ft.  and  7  ft.  above 
the  baseboards,  if  the  area  of  the  openings  is  equivalent  to 
the  surface  of  2  "single  rolls"  of  paper  ? 

5.  Find  the  cost,  at  25^  a  square  yard,  of  plastering  the 
walls  of  a  room  that  is  48  ft.  by  60  ft.  and  18  feet  high, 
deducting  30  square  yards  for  openings. 


PART   III.  377 

Denominate  Numbers. 
336.  Faem  Pkoblems. 

Find  how  many  acres  in — 

1.  A  piece  of  land  1  rod  by  160  rods. 

2.  A  piece  of  land  7  rods  by  160  rods. 

3.  A  piece  of  land  13  rods  by  160  rods. 

4.  A  piece  of  land  22  feet  by  160  rods. 

5.  A  piece  of  land  8^  yards  by  160  rods, 
(a)  Find  the  sum  of  the  five  results. 

6.  A  piece  of  land  8  rods  by  80  rods. 

7.  A  piece  of  land  17  rods  by  80  rods. 

8.  A  piece  of  land  37^  rods  by  80  rods. 

9.  A  piece  of  land  618f  feet  by  80  rods. 

10.  A  piece  of  land  550  yards  by  80  rods. 

(b)  Find  the  sum  of  the  five  results. 

11.  A  piece  of  land  12  rods  by  40  rods. 

12.  A  piece  of  land  27  rods  by  40  rods. 

13.  A  piece  of  land  46  rods  by  20  rods. 

14.  A  piece  of  land  36  rods  by  20  rods. 

15.  A  piece  of  land  264  feet  by  20  rods. 

(c)  Find  the  sum  of  the  five  results. 

16.  A  piece  of  land  1  rod  by  1  mile. 

17.  A  piece  of  land  11  rods  by  1  mile. 

18.  A  piece  of  land  66  feet  by  1  mile. 

19.  A  piece  of  land  99  yards  by  1  mile. 

20.  A  piece  of  land  198  feet  by  ^  of  a  mile. 

(d)  Find  the  sum  of  the  five  results. 

21.  A  piece  of  land  |^  of  a  mile  long  and  as  wide  as  the 
schoolroom. 


378  COMPLETE    ARITHMETIC. 

Denominate  Numbers. 
337.  Faem  Problems. 

1.  A  piece  of  land  1  foot  wide  and  43560  feet  long  is 
how  many  acres  ? 

2.  Change  43560  feet  to  miles. 

3.  A  piece  of  land  1  foot  wide  must  be  how  many  miles 
in  length  to  contain  1  acre  ? 

4.  Some  country  roads  are  66  feet  wide.  How  many 
acres  in  8^  miles  of  such  road  ? 

5.  How  many  acres  in  1  mile  of  road  that  is  4  rods  wide  ? 

6.  A  farmer  walking  behind  a  plow  that  makes  a  furrow 
1  foot  wide  will  travel  how  far  in  plowing  1  acre  ? 

7.  A  farmer  walking  behind  a  plow  that  makes  a  furrow 
16  inches  wide  will  travel  how  far  in  plowing  1  acre  ? 

8.  If  a  mowing  machine  cuts  a  swath  that  averages  4 
feet  in  width,  how  far  does  it  move  in  cutting  1  acre  ? 

9.  If  potatoes  are  planted  in  rows  that  are  3  feet  apart, 
(a)  how  many  miles  of  row  to  each  acre  ?  (b)  How  many 
rods  of  row  to  each  acre  ?  (c)  If  4  rods  of  row  on  the 
average  yield  1  bushel,  what  is  the  yield  per  acre  ? 

10.  Strawberry  plants  are  set  in  rows  that  are  2  feet  apart, 
(a)  How  many  miles  of  row  to  the  acre  ?  (b)  How  many 
rods  of  row  to  the  acre  ?  (c)  How  many  feet  of  row  to  the 
acre  ? 

11.  If  corn  is  planted  in  rows  3^  feet  apart  and  if  the 
"  hills  "  are  3|-  feet  apart  in  the  row,  how  many  hills  to  each 
acre  ? 


PART  III.  379 

Geometry. 
338.  To  Find  the  Area  of  a  Circle. 


1.  Cut  one  half  of  a  circular  piece  of  paper  as  indicated  in 
the  diagram. 

Observe  that  if  the  circle  is  cut  into  a  very  large  number  of  parts 
and  opened  as  shown  in  the  figure,  the  circumference  of  the  circle 
becomes,  practically,  a  straight  line. 

Note. — Imagine  the  circle  cut  into  an  infinite  number  of  parts 
and  thus  opened  and  the  circumference  to  be  a  straight  line. 

Observe  that  a  circle  may  be  regarded  as  made  up  of  an  infinite 
number  of  triangles  whose  united  bases  equal  the  circumference  and 
whose  altitude  equals  the  radius.  Hence  to  find  the  area  of  a  circle 
we  have  the  following : 

EuLE  I.  Multiply  the  circumference  hy  ^  of  the  diameter. 

2.  It  has  already  been  stated  that  if  the  diameter  of  a 
circle  is  1,  its  circumference  is  3.141592.  Hence  the  area 
of  a  circle  whose  diameter  is  1  is  (3.141592  x  |)  .785398. 

3.  A  circle  whose  diameter  is  2,  is  4  times  as  large  as  a 
circle  whose  diameter  is  1 ;  a  circle  whose  diameter  is  3,  9 
times  as  large,  etc.  Hence  to  find  the  area  of  a  circle  we 
have  also  the  following : 

EuLE  II.     Multiply  the  square  of  the  diameter  hy  .785398. 

4.  The  approximate  area  may  be  found  by  taking  f  (or 
.78)  of  the  square  of  the  diameter.      (See  Note  9,  p.  445.) 


380  X)OMPLETE    ARITHMETIC. 

339.  Miscellaneous  Problems. 

1.  Find  the  approximate  area  of  a  circle  whose  diameter 
is  20  feet. 

2.  What  is  the  area  of  a  circle  whose  diameter  is  1  foot  ? 

1  yard  ?     1  rod  ?     1  mile  ? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  2  feet  ? 

2  yards  ?     2  rods  ?     2  miles  ? 

4.  A  horse  is  so  fastened  with  a  rope  halter  that  he  can 
feed  over  a  circle  forty  ieet  in  diameter.  Does  he  feed  over 
more  or  less  than  5  square  rods  ? 

5.  Find  the  approximate  length  (in  rods)  of  the  side  of 
a  square  containing  1  acre. 

6.  Find  the  approximate  diameter  (in  rods)  of  a  circle 
whose  circumference  is  one  mile. 

7.  Find  the  approximate  area  of  the  circle  described  in 
problem  6. 

8.  Find  the  approximate  circumference  of  a  circle  whose 
diameter  is  30  rods. 

9.  The  expression  "a  bicycle  geared  to  68"  means  that 
the  machine  is  so  geared  that  it  will  move  forward  at  each 
revolution  of  the  pedal  shaft  as  far  as  a  68-inch  wheel  would 
move  forward  at  one  revolution.  How  far  does  a  bicycle 
"geared  to  68"  move  forward  at  each  revolution  of  the  pedal 
shaft  ?     A  bicycle  "geared  to  70"  ? 

10.  What  is  the  approximate  circumference  of  the  largest 
circle  that  can  be  drawn  on  the  floor  of  a  room  40  ft.  by  40 
ft.  if  at  its  nearest  points  the  circumference  is  2  feet  from 
the  edge  of  the  floor  ? 


DENOMINATE   NUMBERS. 

Volume  Measure. 

340.  The  standard  unit  of  volume  measure  is  a  cubic  yard, 
which  is  the  equivalent  of  a  1-yard  cube.  This  unit,  like 
the  cubic  foot  and  the  cubic  inch,  is  derived  from  the 
corresponding  unit  of  linear  measure. 

Cubic  Measure. 
1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

Exercise. 

1.  Show  by  a  drawing  that  there  are  27  cu.  ft.  in  a  1-yaid 
cube. 

2.  How  many  cubic  inches  in  1  half  of  a  cubic  foot  ? 

3.  How  many  cubic  inches  in  a  ^-foot  cube  ? 

4.  How  many  cubic  feet  in  1  third  of  a  cubic  yard  ? 

5.  How  many  cubic  feet  in  a  ^-yard  cube  ? 

6.  Estimate  in  cubic  feet  the  amount  of  air  in  the  school- 
room. 

7.  Estimate  in  cubic  yards  the  amount  of  air  in  the 
schoolroom. 

8.  Estimate  in  cubic  inches  the  capacity  of  your  dinner 
box. 

9.  Estimate  in  cubic  feet  the  capacity  of  some  wagon  box. 
10.  Estimate  in  cubic  inches  the  volume  of  the  school 

globe.* 

*  A  globe  is  a  little  more  than  ^  of  the  smallest  cube  from  which  it  could  have 
been  made.    See  note  10,  p.  445. 

381 


382  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Volume  Measure. 

341.  Wood  is  usually  measured  by  the  cord.  A  cord  is  a 
pile  4  feet  wide,  4  feet  high,  and  8  feet  long,  or  its  equiva- 
lent.    Hence — 

128  cubic  feet  =  1  cord. 

Problems. 

1.  Estimate  the  number  of  cords  of  wood  that  could  be 
put  upon  the  floor  of  the  school  room  if  the  desks  were  re- 
moved and  the  wood  piled  to  the  depth  of  four  feet. 

2.  If  4-foot  wood  is  piled  6  feet  high  what  must  be  the 
length  of  the  pile  to  contain  100  cords? 

3.  How  many  cords  of  wood  in  a  pile  8  feet  wide,  8  feet 
high,  and  1 6  feet  long  ? 

4.  Compare  the  amount  of  wood  in  the  pile  described  in 
problem  3,  with  the  amount  in  a  pile  one  half  as  wide,  one 
half  as  high,  and  one  half  as  long. 

5.  If  I  pay  $1.10  a  cord  for  sawing  wood,  cutting  each 
4-foot  stick  into  3  pieces,  how  much  ought  I  to  pay  for  cut- 
ting each  4-foot  stick  into  4  pieces  ? 

6.  A  pile  of  wood  4  ft.  high,  4  ft.  wide,  and  192  ft.  long 

contains    cords.      How  many  cords  in  a  pile  4  feet 

high,  192  feet  long,^  and  46  inches  wide  ? 

7.  A  pile  of  wood  is  as  wide  as  it  is  high  and  32  feet 
long.  It  contains  9  cords.  What  is  the  width  and  height 
of  the  pile  ? 

8.  How  many  cords  of  4-foot  wood  can  be  piled  in  a  cellar 
that  is  24  feet  wide  and  32  feet  long,  provided  the  pile  is  4 
feet  high  and  one  end  of  each  4-foot  stick  touches  a  wall  of 
the  cellar  ? 


PART  III.  383 

Denominate  Numbers— Volume  Measure. 

342.  Rough  Stone  is  usually  measured  by  the  cord.  A 
pile  4  feet  high,  4  feet  wide,  and  8  feet  long  or  its  equiv- 
alent, is  1  cord. 

Note. — One  cord  of  good  stone  is  sufficient  for  about  100  cubic 
feet  of  wall.  Hence  in  estimates  it  is  customary  to  use  the  number 
100  instead  of  128 ;  thai  is,  as  many  cords  of  stone  will  be  required 
for  a  given  wall  as  100  cubic  feet  is  contained  times  in  the  number 
of  cubic  feet  in  the  wall. 

Problems. 

1.  Estimate  the  number  of  cords  of  stone  necessary  for  a 
cellar  wall  18  inches  thick,  the  inside  dimensions  of  the 
cellar  being  15  feet  by  18  feet  and  7  feet  deep,  no  allowance 
being  made  for  openings  in  the  wall. 

2.  What  are  the  outside  dimensions  of  the  wall  of  the 
cellar  described  in  problem  1  ? 

3.  What  length  of  wall  7  feet  high  and  18  inches  thick  is 
equivalent,  so  far  as  amount  of  stone  is  concerned,  to  the 
cellar  wall  described  in  problem  1  ? 

4.  If  ^  of  the  depth  of  the  cellar  described  above  is  to  be 
below  the  surface  of  the  ground,  how  many  cubic  yards  of 
earth  must  be  excavated  ? 

5.  How  many  per  cent  less  of  stone  will  be  required  for  a 
16-inch  wall  than  for  an  18-inch  wall  ? 

6.  Estimate  the  stone  necessary  for  a  wall  100  yards  long, 
1 1  feet  high,  and  2  feet  thick. 

7.  If  the  specific  gravity  of  stone  is  2|^  and  each  cord  is 
equivalent  to  100  solid  feet,  how  much  does  a  cord  of  stone 
weigh  ? 

8.  If  the  specific  gravity  of  a  certain  stone  is  2i,  what  is 
the  weight  of  a  block  8  feet  by  2  feet  by  2  feet  ? 


384  COMPLETE    ARITHMETIC. 

Denominate  Numbers — Volume  Measure. 

343.  An  ordinary  brick  is  2  in.  by  4  in.  by  8  in.  and 
weighs  about  4  pounds. 

Problems. 

1.  How  many  bricks  are  equivalent  to  1  cubic  foot  ? 

Note. — When  bricks  are  laid  in  mortar  in  the  usual  way,  about 
22  bricks  are  required  to  make  a  cubic  foot  of  wall. 

2.  Estimate  the  number  of  bricks  necessary  for  a  cellar 
wall  12  inches  thick,  the  inside  dimensions  of  the  cellar 
being  15  feet  by  18  feet,  and  7  feet  deep,  no  allowance  being 
made  for  openings  in  the  wall  ? 

3.  What  are  the  outside  dimensions  of  the  wall  of  the 
cellar  described  in  problem  2  ? 

4.  What  length  of  wall  7  feet  high  and  12  inches  thick 
is  equivalent,  so  far  as  the  number  of  bricks  required  is 
concerned,  to  the  cellar  wall  described  in  problem  2  ? 

5.  If  f  of  the  depth  of  the  cellar  described  above  is  to  be 
below  the  surface,  how  many  cubic  yards  of  earth  must  be 
excavated  ? 

6.  Estimate  the  number  of  bricks  necessary  for  a  wall  100 
yards  long,  11  feet  high,  and  1  foot  thick. 

7.  If  a  brick  is  exactly  2  in.  by  4  in.  by  8  in.  and  weighs 
exactly  4^  lbs.  what  is  its  specific  gravity  ?     (Note  i6,  p.  446.) 

8.  Find  the  approximate  weight  (in  tons)  of  a  pile  of 
bricks  as  long  as  your  school-room,  2  feet  wide,  and  4  feet 
high. 

9.  Find  the  approximate  weight  of  a  chimney,  outside 
dimensions,  16  in.  by  16  in.,  and  20  ft.  high,  the  flue  being 
8  in.  by  8  in. 


PART  III.  385 

Denominate  Numbers — Lumber. 

344.  A  foot  of  lumber  is  a  board  1  foot  square  and  1  inch 
thick  or  its  equivalent.     (Note  ii,  p.  445.) 

Note  1. — An  exception  to  the  foregoing  is  made  in  the  measure- 
ment of  boards  less  than  1  inch  in  thickness.  A  square  foot  of  such 
boards  is  regarded  as  a  foot  of  lumber,  whatever  the  thickness. 

Exercise. 

Tell  the  number  of  feet  of  lumber  in  each  of  the  following 
boards,  the  thickness  in  each  case  being  one  inch  (or  less) : 
1  in.  wide  and  12  ft.  long.  2  in.  wide  and  12  ft.  long. 
3  in.  wide  and  12  ft.  long.  4  in.  wide  and  12  ft.  long. 
7  in.  wide  and  12  ft.  long.  13  in.  wide  and  12  ft.  long. 
9  in.  wide  and  12  ft.  long.     12  in.  wide  and  12  ft.  long. 

(a)  How  many  feet  (of  lumber)  in  the  eight  boards  ? 

Problems. 

1.  How  much  lumber  in  6,  12 -ft.,  1-in.  boards  whose 
widths  are  11  in.,  13  in.,  9  in.,  10  in.,  12  in.,  and  14  in.? 

2.  How  much  lumber  in  5,  12-ft.,  |--in.  boards  whose 
widths  are  10  in.,  12  in.,  12  in.,  11  in.,  and  14  in.? 

3.  How  much  lumber  in  7,  12-ft.,  |^-in.  boards  whose 
widths  are  9  in.,  8  in.,  5  in.,  7  in.,  8  in.,  6  in.,  and  9  in.  ? 

4.  How  much  lumber  in  8,  12-ft.,  1-in.  boards  each  of 
which  is  12  inches  wide  ? 

5.  How  much  lumber  in  54,  12-ft.,  1-in.  boards  each  of 
which  is  6  inches  wide  ? 

(b)  Find  the  sum  of  the  five  results. 


386  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Lumber. 

Problems. 

Note  2. — A  14-foot  board  contains  ^  more  lumber  than  a  12-ft. 
board  of  the  same  width  and  thickness.  Hence  to  find  the  number 
of  feet  of  lumber  in  14-foot  boards,  find  the  number  of  feet  in  as 
many  12-foot  boards  and  add  to  the  result  ^  of  itself.* 

1.  How  much  lumber  in  5,  14-ft.,  1-in.  boards  whose 
widths  are  11  in.,  12  in.,  12  in.,  15  in.,  and  10  in.  ? 

2.  How  much  lumber  in  a  pile  of  14-ft.  boards  whose 
united  width  is  8  feet  7  inches  ? 

3.  How  much  lumber  in  56,  14-ft.  boards  each  of  which 
is  6  inches  wide  ?  f 

4.  How  much  lumber  in  24,  14-ft.  boards  each  of  which 
is  12  inches  wide  ? 

(a)  Find  the  sum  of  the  four  results. 

Problems. 

Note  3. — A  16-foot  board  contains  |  more  lumber  than  a  12-ft. 
board  of  the  same  width  and  thickness.  Make  a  rule  for  finding 
the  number  of  feet  of  lumber  in  16-foot  boards. 

1.  How  much  lumber  in  5,  16-ft.,  1-in.  boards  whose 
widths  are  12  in.,  10  in.,  14  in.,  13  in.,  and  12  in.  ? 

2.  How  much  lumber  in  a  pile  of  16-foot  boards  whose 
united  width  is  9  feet  8  inches  ? 

3.  How  much  lumber  in  48,  16-ft.  boards  each  of  which 
is  6  inches  wide  ? 

4.  How  much  lumber  in  34,  16-ft.  boards  each  of  which 
is  12  inches  wide  ? 

(b)  Find  the  sum  of  the  four  results. 

*  Take  the  nearest  integral  number  of  feet. 

I  How  much  lumber  in  one  14-foot  board  6  inches  wide? 


PART  III.  387 

Denominate  Numbers— Lumber. 

Problems. 

Note  4. — A  Ij-incli  board  contains  i  more  lumber  than  a  1-inch 
board  of  the  same  width  and  length.  A  l|-inch  board  contains  J 
more  lumber  than  a  1-inch  board  of  the  same  width  and  length. 

1.  How  much  lumber  in  4,  12-foot,  IJ-in.  boards  whose 
widths  are  12  in.,  13  in.,  14  in.,  and  13  in.  ? 

2.  How  much  lumber  in  4,  16-foot,  1^-in.  boards  whose 
widths  are  13  in.,  16  in.,  12  in.,  and  13  in.  ? 

3.  How  much  lumber  in  4,  18 -foot,  l|^-in.  boards,  each  of 
which  is  12  inches  wide? 

4.  How  much  lumber  in  4,  16-ft.,  1^-in.  boards,  each  of 
which  is  6  inches  wide  ? 

(a)  Find  the  sum  of  the  four  results. 

Problems. 

Note  5. — A  "2  by  4,  12"  is  a  piece  of  lumber  2  in.  thick,  4  in. 
wide,  and  12  feet  long. 

Find  the  number  of  feet  of  lumber  in  each  of  the  follow- 
ing items : 

1.  16  pieces  2x4,  12. 

2.  18  pieces  4x4,  12. 

3.  25  pieces  2x8,  12. 

4.  30  pieces  2x6,  12. 

5.  20  pieces  4  x  6,  12. 

(b)  Find  the  sum  of  the  five  results. 

Observe  that  in  a  12-foot  piece  of  lumber  there  are  as  many  feet  as 
there  are  square  inches  in  the  cross-section.  A  piece  of  lumber  1  in. 
by  1  in.  and  12  feet  long  is  1  foot  of  lumber;  a  piece  2  in.  by  2  in. 
is  4  feet  of  lumber ;  a  piece  2  in  by  3  in.  is  6  feet  of  lumber,  etc. 


388  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Lumber. 

Pkoblems. 

Note  6. — In  the  measurement  of  timbers  of  all  sizes  it  is  custom- 
ary to  consider  each  piece  as  containing  the  integral  number  of  feet 
nearest  to  the  actual  content.  Thus,  a  piece  of  2  x  4,  14,  actually 
contains  9|  feet,  but  in  all  lumber  yards  it  is  counted  as  9  feet.  A 
piece  of  2  X  4,  16,  actually  contains  10|  feet,  but  it  is  counted  as  11 
feet. 

Find  the  number  of  feet  of  lumber  in  each  of  the  following 
items  : 

1.  16  pieces  2x4,  14. 

2.  24  pieces  4x4,  14. 

3.  32  pieces  2x8,  14. 

4.  17  pieces  4  x    6,  14. 

5.  15  pieces  8  x    8,  16. 

6.  12  pieces  4  x  10,  16. 

7.  14  pieces  8  x  12,  16. 

8.  6  pieces  12  x  12,  24. 

(a)  Find  the  sum  of  the  eight  results. 

Pkoblems. 

Note  7. — "  Lumber  at  $15  per  M,"  means  that  the  lumber  is  sold 
at  the  rate  of  $15  per  1000  feet. 

Find  the  cost : 

1.  26,  16-foot,  6-in.  fence  boards  @  $15  per  M. 


2.  34,  14-foot,  12-in.  stock  boards 

3.  20  pieces  2x4,  16, 

4.  14  pieces  4  x  6,  18, 

5.  25  pieces  4  x  6,  16, 

6.  18  pieces  4x4,  14, 

(b)  Find  the  sum  of  the  six  results. 


$18  per  M. 
$16  per  M. 
$16  per  M. 
$15  per  M. 

$15  per  M. 


PART  III.  389 

Qeometry. 

345.  To  Find  the  Solid  Content  of  a  cylinder  or  of  a 
right  prism.* 

Observe  that  in  any  cylinder  or  right  prism  the 
number  of  cubic  units  in  one  layer  1  unit  high  (as 
indicated  in  the  diagrams)  is  equal  to  the  number 
of  square  units  in  the  area  of  the  base.     Thus,  if 
there  are  4i  square  units  in  the  area  of  the  base 
there  are  4|^  cubic  units  in  one  layer.    The  content 
of  the  entire  solid  is  as  many  times  the  cubic 
units  in  one  layer,  as  the  solid  is  linear  units  in 
height.      Hence    the    rule    as    usually    given: 
"  Multiply  the  area  of  the  base  hy  the  altitude." 

To  THE  Teacher.  — This  rule  must  be  carefully 
interpreted  by  the  pupil.  He  must  not  be  allowed  the 
misconception  that  area  multiplied  by  any  number  can 
give  solid  content,  except  through  such  interpretation  as 
is  suggested  in  the  above  observation.      (Note  12,  p.  446.) 


Problems. 

1.  Find  the  solid  content  of  a  square  right  prism  whose 
base  is  6  in.  by  6  in.,  and  whose  altitude  is  8  inches. 

2.  Find  the  approximate  solid  content  of  a  cylinder  6 
inches  in  diameter  and  10  inches  long. 

3.  Find  the  solid  content  of  a  triangular  prism  the  area 
of  whose  base  is  15  sq.  in.,  and  whose  altitude  is  11|-  inches. 

4.  Find  the  solid  content  of  an  hexagonal  prism  the  area 
of  whose  base  is  18  inches,  and  whose  altitude  is  10|-  inches- 

*  A  right  prism  is  a  solid  wiiose  bases,  or  ends,  are  similar,  equal,  and  parallel 
plane  polygons,  and  whose  lateral  faces  are  perpendicular  to  its  bases. 


390  COMPLETE    ARITHMETIC. 

346.  Miscellaneous  Problems. 

1.  Find  the  solid  content  of  an  octagonal  right  prism  the 
area  of  whose  base  is  24  square  inches,  and  whose  altitude 
is  15  inches. 

2.  What  is  the  solid  content  of  a  cylinder,  or  of  any  right 
prism,  the  area  of  whose  base  is  30  square  inches,  and  whose 
altitude  is  1 2  inches  ? 

3.  How  many  cubic  feet  of  earth  must  be  removed  to  dig 
a  well  6  feet  in  diameter  and  20  feet  deep  ?  * 

4.  Find  the  approximate  number  of  feet  of  H-m.  lumber 
required  to  make  the  lining  of  the  sides  of  a  cylindrical  silo 
that  is  20  feet  in  diameter  and  30  feet  deep. 

5.  Find  the  approximate  number  of  cords  of  rough  stone  in 
a  cylindrical  pile  that  is  1 6  feet  in  diameter  and  six  feet  deep. 

6.  Find  the  approximate  number  of  brick  necessary  for  a 
solid  cylindrical  foundation  that  is  9  feet  in  diameter  and  4 
feet  high. 

7.  If  the  average  specific  gravity  of  the  brick  and  mortar 
used  in  the  foundation  described  in  problem  6,  is  1.9,  how 
much  does  the  entire  foundation  weigh  ? 

8.  Find  the  weight  in  kilograms  of  a  column  of  water  1 
decimeter  square  and  10  meters  deep. 

9.  Find  the  weight  in  pounds  of  1000  feet  of  white  pine 
1-inch  boards,  the  specific  gravity  being  .6. 

10.  Find  the  weight  of  a  load  (1  cubic  yard)  of  wet  sand, 
the  specific  gravity  being  exactly  2. 

*  The  exact  number  of  cubic  feet  cannot  be  expressed  in  figures.  An  approxi- 
mation that  will  answer  many  practical  purposes  may  be  obtained  by  regarding  the 
circle  (base)  as  3  of  its  circumscribed  square.  If  an  answer  more  nearly  accurate  is 
required  use  .78  instead  of  2. 


DENOMINATE   NUMBEES. 

Capacity. 

347.  The  standard  unit  of  capacity  used  in  measuring 
liquids  is  a  gallon.     A  gallon  equals  231  cubic  inches. 

Liquid  Measure. 

4  giUs  (gi.)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts        =  1  gallon  (gal.). 
31^  gallons  =:  1  barrel  (bbl.). 

Observe  that  1  cubic  foot  =  nearly  7 1  gallons. 
Observe  that  4.2  cubic  feet  =  nearly  1  barrel. 
A  kerosene  barrel  contains  about  52  gallons.     It  equals  nearly  7 
cubic  feet. 

Problems. 

1.  Find  the  capacity  (approximate  or  exact),  in  gallons,  of 
a  rectangular  tank  3  ft.  by  4  ft.  by  8.  ft. 

2.  Find  the  approximate  capacity,  in  gallons,  of  a  cylin- 
drical tank  4  feet  in  diameter  and  4  feet  deep. 

3.  Find  the  approximate  capacity,  in  barrels  (31|-  gal.),  of 
a  rectangular  tank  2  ft.  by  4  ft.  by  12  ft. 

4.  Find  the  approximate  capacity,  in  barrels  (31^  gal.),  of 
a  cylindrical  cistern  6  ft.  in  diameter  and  6  ft.  deep. 

5.  Find  the  approximate  capacity,  in  barrels  (31^  gal.),  of 
a  cylindrical  cistern  12  ft.  in  diameter  and  6  ft.  deep. 

6.  Find  the  approximate  capacity,  in  barrels  (31|^  gal.),  of 
a  cylindrical  cistern  12  ft.  in  diameter  and  12  ft.  deep. 

391 


392  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Capacity. 

348.  The  standard  unit  of  capacity  used  in  measuring 
grain,  fruits,  vegetables,  lime,  coal,  etc.,  is  a  bushel.  A 
bushel  equals  2150.4  cubic  inches. 

Note. — In  measuring  large  fruits,  vegetables,  lime,  and  coal,  the 
unit  is  the  "heaped  busliel."  A  heaped  bushel  equals  about  11 
"  stricken  bushels." 

Dry  Measure. 

2  pints  (pt.)   =  1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.). 

A  bushel  is  nearly  IJ  cubic  feet. 
A  "  heaped  bushel "  is  about  1 J  cubic  feet. 
A  "dry  gallon"  (4  quarts  dry  measure)  equals  268.8  cubic  inches. 

Enough  "ear  corn"  to  make,  when  shelled,  one  bushel,  occupies 
about  21  cubic  feet.  If  the  corn  is  inferior  in  quality  it  will  occupy 
more  space  than  this — sometimes  2^  cubic  feet. 

Problems. 

1.  Find  the  capacity  in  bushels  of  a  wheat  bin  8  ft.  by  8 
ft.  by  10  ft.* 

2.  Give  the  dimensions  of  the  smallest  bin  in  which  1000 
bushels  of  oats  may  be  stored. 

3.  In  a  bin  12  feet  square,  there  is  rye  to  the  depth  of  7|- 
feet.     How  many  bushels  ? 

4.  How  many  bushels  of  potatoes  (without  heaping  the 
bin)  may  be  stored  in  a  bin  that  is  8  ft.  by  4  ft.  by  6  ft.? 

5.  If  the  com  is  of  excellent  quality,  how  many  bushels 
of  "  shelled  corn  "  may  be  expected  from  a  crib  of  ear  corn, 
8  ft.  by  10  ft.  by  80  ft.? 

*For  many  practical  purposes  the  approximate  ratio  (li)  of  the  bushel  to  the 
cubic  foot  will  give  in  such  problems  as  these,  results  sufficiently,  accurate. 


PART    III. 


393 


Denominate  Numbers— Weight. 

349.  The  standard  unit  of  weight  in  common  use  is  a 
pound  Avoirdupois. 

Avoirdupois  Weight. 

16  ounces  (oz.)   =  1  pound  (lb.). 
2000  pounds       =  1  ton  (T.). 
The  abbreviation  for  1  hundredweight  (100  lb.)  is  cwt. 


Miscellaneous  Weights. 


1  gallon  of  water 

1  gallon  of  milk 

1  gallon  of  kerosene 

1  cubic  foot  of  water 

1  bushel  of  wheat 

1  bushel  of  beans 

1  bushel  of  clover  seed 

1  bushel  of  potatoes 

1  bushel  of  shelled  corn 

1  bushel  of  ear  corn 

1  bushel  of  rye 

1  bushel  of  barley 

1  bushel  of  oats 

1  barrel  of  flour 

1  barrel  of  beef  or  pork 


=  about  8^  lb. 

=  about  8.6  lb. 

=  about  6^  lb. 

=  62^  lb. 

=  60  lb. 

=  60  lb. 

=  60  lb. 

=  60  lb. 

=  56  lb. 

=  70  lb.* 

=  56  lb. 

=  48  lb. 

=  32  lb. 

=  196  lb. 

=  200  lb. 


Problems. 
Find  the  cost — 

1.  Of  2650  lb.  coal  at  S5.50  per  ton. 

2.  Of  2650  lb.  oats  at  24^  a  bushel. 

3.  Of  3330  lb.  wheat  at  80^  a  bushel. 

4.  Of  4650  lb.  potatoes  at  42^  a  busheL 
(a)  Find  the  sum  of  the  four  results. 

*This  means  the  amount  of  ear  corn  required  to  make  1  bushel  of  shelled  corn. 


394  COMPLETE    ARITHMETIC. 

Denominate  Numbers — Weight. 

Problems. 
Find  the  cost — 

1.  Of  2560  lb.  hay  at  $7.50  per  ton. 

2.  Of  1430  lb.  straw  at  30^  per  cwt. 

3.  Of  2|-  tons  meal  at  f  of  a  cent  a  pound. 

4.  Of  1|-  tons  corn  husks  at  1^  cents  a  pound. 

5.  Of  3420  lb.  hay  at  $8.00  per  ton. 
(a)  Find  the  sum  of  the  five  results. 

Find  the  cost — 

6.  Of  2140  lb.  oats  at  24^  a  bushel. 

7.  Of  2140  lb.  corn  at  2^  a  bushel. 

8.  Of  2140  lb.  wheat  at  90^  a  bushel. 

9.  Of  2140  lb.  barley  at  36^  a  bushel. 

10.  Of  2140  lb.  rye  at  42^  a  bushel. 

(b)  Find  the  sum  of  the  five  results. 

Find  the  cost — 

11.  Of  520  lb.  clover  seed  at  $6.30  a  bushel. 

12.  Of  520  lb.  potatoes  at  75^  a  bushel. 

13.  Of  520  lb.  beans  at  $2.15  a  bushel. 

14.  Of  520  lb.  corn  at  20^  a  bushel. 

15.  Of  520  lb.  ear  corn  at  35^  a  bushel. 

(c)  Find  the  sum  of  the  five  results. 

Find  the  approximate  weight — 

16.  Of  a  barrel  of  kerosene. 

17.  Of  1  quart  of  milk. 

18.  Of  the  oats  that  will  fill  a  bin  that  is  4  ft.  by  4  ft.  by 
9  ft. 

19.  Of  the  water  that  will  fill  a  taxxk  that  is  2  ft.  by  2  ft. 
by  12  ft. 


PART    III.  395 

Denominate  Numbers— Weight. 

350.  Troy  weight  is  used  in  weighing  gold,  silver,  and 
jewels. 

Troy  Weight. 

24  grains  (gr.)     =  1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

351.  Apothecaries'  weight  is  used  in  mixing  medicines 
and  in  selling  them  at  retail. 

Apothecaries'  Weight. 

20  grains  (gr.)  =  1  scruple  (3). 
3  scruples  =  1  dram  (  3  ). 

8  drams  =  1  ounce  (  §  ). 

12  ounces  =  1  pound  (Bb). 

Note  1. — The  pound  Troy  and  the  pound  Apothecary  are  equal, 
each  weighing  5760  grains.  The  pound  Avoirdupois  weighs  7000 
Troy  or  Apothecary  grains. 

Note  2. — The  ounce  Troy  and  the  ounce  Apothecary  are  each 
480  grains;  the  ounce  Avoirdupois  is  437 1  grains. 

Query. — Which  is  heavier,  a  pound  of  feathers  or  a  pound  of 
gold  ?     An  ounce  of  feathers  or  an  ounce  of  gold  ? 

Problems. 

1.  Change  5  lb.  Avoirdupois  weight  to  pounds,  ounces, 
etc.,  Apothecaries'  weight. 

2.  How  many  5  gr.  powders  can  be  made  from  one  Avoir- 
dupois ounce  of  quinine  ? 

3.  One  Avoirdupois  ton  of  gold  is  how  many  Troy  pounds  ? 

4.  Twenty-four  Troy  pounds  equal  how  many  Avoirdupois 
pounds  ? 


396  COMPLETE    ARITHMETIC. 

Denominate  Numbers — Time. 

352.  The  standard  units  in  the  measurement  of  time  are 
the  day  and  the  year. 

Note  1. — The  solar  day  is  the  interval  between  the  time  when 
the  sun  is  on  a  given  meridian  and  the  time  when  it  appears  on  that 
meridian  again.  These  intervals  (solar  days)  are  not  perfectly  uni- 
form. The  average  of  these  intervals  is  the  mean  solar  day — the 
day  noted  by  our  watches  and  our  clocks— the  day,  one  twenty-fourth 
of  which  is  called  an  hour. 

Note  2.— The  solar  year  is  the  time  of  one  revolution  of  the 
earth  around  the  sun,  or  nearly  365 i  mean  solar  days.  The  calendar 
year  of  365  days  is  nearly  6  hours  less  than  the  solar  year.  Four 
years  of  365  days  each,  would  lack  nearly  24  hours  (4  times  J  da.) 
of  being  equal  to  4  solar  years.  Hence  1  day  is  added  to  365  every 
fourth  year,  ^ith  the  exception  noted  below. 

Note  3. — The  exact  length  of  a  solar  year  is  365  da.  5  hr.  48  min. 
48  sec.  Since  this  lacks  11  min.  12  sec.  of  being  365^  days,  it  fol- 
lows that  if  every  fourth  year  should  contain  366  days,  in  400  years 
the  calendar  years  would  amount  to  3  days  (400  times  11  min.  12 
sec.)  more  than  the  solar  years.  Hence  three  of  the  years  that 
would  otherwise  contain  366  days,  are  made  to  contain  365  days. 
The  years  thus  changed  to  365  days  are  those  ending  the  centuries 
(1800,  1900,  2000,  2100,  etc.)  unless  the  number  denoting  the  year 
is  exactly  divisible  by  400. 

Measure  of  Time. 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =  1  hour  (hr.). 

24  hours  —  1  day  (da.). 

7  days  =  1  week  (wk.). 

365  days  =  1  common  year. 
52  wk.  1  da.  =  1  common  year. 

366  days  —  1  leap  year. 
52  wk.  2  da.  =1  leap  year. 
12  months  =  1  year. 


PART   III.  397 

Denominate  Numbers— Time. 

Exercises. 

1.  If  Jan.  1  of  a  common  year  is  Monday,  Feb.  1  is  ; 

March  1  is  ;  April  1  is  ;  May  1  is  ;  June  1 

is  ;  July  1  is  ;  August  1  is  ;  September  1  is 

;  October  1  is  ;  November  1  is  ;  December  1 

is  . 


2.  Jan.  1,  1899,  was  Sunday.  Tell  the  day  of  the  week 
for  each  of  the  following  dates : 

1900,  January  1 ;  February  1 ;  February  8. 

1901,  January  1 ;  February  1 ;  February  9. 

1902,  January  1 ;  February  1 ;  March  1. 

1903,  January  1 ;  February  1 ;  March  10. 

Problems. 

1.  How  many  days  from  Jan.  1,  1900,  to  Aug.  17,  19001 

2.  Jan.  1,  1900,  falls  on  Monday.  Upon  what  day  of  the 
week  does  Aug.  17,  1900,  fall? 

3.  What  month  begins  on  the  same  day  of  the  week  as 
January  in  every  common  year  ? 

4.  What  months  begin  on  the  same  day  of  the  week  as 
February  in  every  common  year  ? 

5.  If  January  begins  on  Sunday,  (a)  how  many  Sundays 
in  the  month  ?  (b)  How  many  Mondays  ?  (c)  How  many 
Tuesdays  ?     (d)  How  many  Wednesdays  ? 

6.  If  a  common  year  begins  on  Sunday,  (a)  how  many 
Sundays  in  the  year  ?     (b)  How  many  Mondays  ? 

7.  (a)  How  many  days  old  are  you?  (b)  How  many 
weeks  old?    (c)  Upon  what  day  of  the  week  were  you  born? 


398  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Circular  Measure. 

353.  For  the  purpose  of  measurement,  every  circumference 
is  supposed  to  be  divided  into  360  equal  parts.  Each  of 
these  parts  is  called  an  arc  of  1  degree. 

354.  An  angle  is  measured  by  regarding  its  vertex  as  the 
center  of  a  circle,  its  sides  being  extended  until  they  cut  the 
circumference.  The  angle  is  measured  by  the  arc  lying 
between  its  sides.  If  the  intercepted  arc  is  an  arc  of  45 
degrees,  the  angle  is  an  angle  of  45  degrees ;  if  the  arc  is  20 
degrees,  the  angle  is  20  degrees,  etc. 

Circular  and  Angular  Measure. 

60  seconds  (")  =:  1  minute  ('). 
60  minutes  =  1  degree  (°). 
360  degrees       =  1  circumference. 

355.  In  geography,  a  meridian  is  a  north  and  south  line 
on  the  surface  of  the  earth,  extending  from  pole  to  pole. 

356.  Longitude  is  distance  east  or  west  in  degrees  (or 
parts  of  degrees)  from  a  given  meridian.  Longitude  is 
usually  measured  from  the  meridian  that  passes  through 
Greenwich. 

Problems. 

Find  difference  in  longitude  between — 

1.  Eome,  12°  27'  east,  and  Washington,  70°  2'  48'' west. 

2.  Washington  and  Chicago,  87°  37'  30"  west. 

3.  Chicago  and  Denver,  104°  59'  23"  west. 

4.  Denver  and  San  Francisco,  122°  24'  15"  west. 

5.  San  Francisco  and  Berlin,  13°  23'  53"  east. 

6.  Washington  and  Honolulu,  157°  50'  36"  west. 

7.  Washington  and  Manila,  121°  east. 


PART   III.  399 

Denominate  Numbers— Longitude  and  Time. 

357.  One  degree  of  longitude  corresponds  to  4  minutes  of 
time. 

Explanatory — The  sun  seems  to  move  over  360  degrees  of 
longitude  in  24  hours ;  over  1  degree  in  ^^-^  of  24  hours  =  4  minutes. 

Problems.* 

1.  When  it  is  noon  at  Greenwich,  what  is  the  time  at 
Washington,  77°  2'  48''? 

2.  When  it  is  noon  at  Washington,  what  is  the  time  at 
Greenwich  ? 

3.  When  it  is  noon  at  Portland,  Maine,  70°  15'  40", 
what  is  the  time  at  San  Francisco,  122°  24'  15"? 

4.  When  it  is  noon  at  San  Francisco,  what  is  the  time  at 
Portland,  Maine  ? 

358.  Standard  Railroad  Time.  Every  railroad  train  in 
North  America  is  run  on  the  time  of  some  one  of  the  follow- 
ing meridians : 

60th  meridian  (passing  through  Labrador). 

75th  meridian  (passing  near  Philadelphia). 

90th  meridian  (passing  near  St.  Louis). 
105th  meridian  (passing  near  Denver). 
120th  meridian  (passing  near  Carson  City). 

Observe  that  each  of  the  above  numbers  after  the  first  is  15  greater 
than  the  one  preceding  it;  and  that  15  degrees  of  longitude  corre- 
spond to  1  hour  of  time. 

*  At  first  the  pupil  should  give  an  approximate  answer  to  these  problems,  consid- 
ering integral  degrees  only.  Later,  if  thought  advisable,  he  may  take  into  the  account 
the  parts  of  degrees.  One  arc  minute  corresponds  to  ^  of  4  minutes  of  time,  or  4 
seconds  of  time.  One  arc  second  corresponds  to  gsW  of  4  minute^  of  time,  or  b%  of 
one  second  of  time.  That  is,  each  arc  degree  corresponds  to  4  minutes  of  time;  each 
arc  minute,  to  4  seconds  of  time;  each  arc  second,  to  Ss  of  a  second  of  time.  Hence, 
multiplying  by  4  the  figures  standing  for  degrees,  minutes,  and  seconds,  of  longitude 
Will  give  the  figures  standing  for  minutes,  seconds,  and  60ths  of  seconds,  of  tim«i. 


400  COMPLETE    ARITHMETIC. 

Denominate  Numbers— Value. 

359.  The  standard  unit  of  value  in  the  United  States  is 
the  dollar. 

United  States  Money. 

10  mills  (m.)  =  1  cent  (ct.  or  ^). 
10  cents  =  1  dime  (d.). 

10  dimes         =  1  dollar  ($). 

360.  The  standard  unit  of  value  in  Great  Britain  and 
Ireland  is  the  pound.  Its  value,  reckoned  in  United  States 
money,  is  $4.8  6 6f 

English,  or  Sterling,  Money. 

4  farthings  (far.)  =  1  penny  (d.). 
12  pence  =  1  shilling  (s.). 

20  shillings  =  1  pound  (£). 

5  shillings  =  1  crown. 

21  shillings  =  1  guinea. 

361.  The  standard  unit  of  value  in  France  is  the  franc. 
Its  value,  reckoned  in  United  States  money,  is  19.3^. 

362.  The  standard  unit  of  value  in  Germany  is  the  mark. 
Its  value,  reckoned  in  United  States  money,  is  23.85^. 

363.  The  standard  unit  of  value  in  Eussia  is  the  ruble. 
Its  value,  reckoned  in  United  States  money,  is  $.772. 

Problems. 

1.  Find  the  value  of  a  guinea  in  United  States  money. 

2.  Find  the  value  of  $1000  in  English  money. 

3.  Find  the  value  of  £5000  in  United  States  money. 

4.  Find  the  value  of  £1000  in  Eussian  money. 

5.  Find  the  value  of  4000  marks  in  English  money. 

6.  Find  the  value  of  8000  francs  in  United  States  money. 


SHOET   METHODS. 

Multiplication  and  Division. 

Art.  1.  To  multiply  a  number  by  50  :     Multiply  ^  of  the 
number  hy  10^.     Why? 

46  X  50         I- of  46  X  100  ^2300 

47  X  50         I  of  47  X  100  =  2350 

Multiply: 


44  by  50 

32  by  50 

35  by  50 

36  by  50 

38  by  50 

27  by  50 

64  by  50 

42  by  50 

55  by  50 

82  by  50 

76  by  50 

43  by  50 

46  by  50 

52  by  50 

53  by  50 

(a)  Find  the  sum  of  the  fifteen  products. 

Art.  2.  To  multiply  a  number  by  51 :    Take  50  times  i 
number,  to  which  add  the  number  itself.     Why  ? 

48  X  50   50  times  48  =  2400   2400  +  48  =  2448 
37  X  51   50  times  37  =  1850   1850  +  37  =  1887 


Multiply : 

26  by  51 

34  by  51 

35  by  51 

46  by  51 

32  by  51 

29  by  51 

24  by  51 

42  by  51 

43  by  51 

66  by  51 

84  by  51 

33  by  51 

36  by  51 

38  by  51 

39  by  51 

(b)  Find  the  sum  of  the  fifteen  products. 

401 


402  COMPLETE   ARITHMETIC. 

Art.  3.  To  multiply  a  number  by  52 :     Take  50  times  the 
number,  to  which  add  tivice  the  number.     Why  ? 

34x52        50  times  34  =  1700       1700  +  68  =  1768 
45x52        50  times  45  =r  2250       2250  +  90  =  2340 

Multiply : 
26  by  52         38  by  52         27  by  52 
18  by  52         14  by  52         17  by  52 
24  by  52         32  by  52         35  by  52 
36  by  52         44  by  52         23  by  52 

(c)  Find  the  sum  of  the  twelve  products. 

Art.  4.  To  multiply  a  number  by  49 :     Take  50  times  the 
number,  from  which  subtract  the  number  itself.     Why  ? 
24x49        50  times  24=  1200       1200-24  =  1176 
33x49        50  times  33  =  1650       1650-33  =  1617 

Multiply : 
18  by  49         22  by  49         27  by  49 
28  by  49         34  by  49         35  by  49 
16  by  49         46  by  49         43  by  49 

(d)  Find  the  sum  of  the  nine  products. 

Art.  5.  To  multiply  a  number  by  33^:  Multiply  ^  of  the 
number  by  100.     Why? 

36  X  33i  I  of  36  X  100  =  1200 

37  X  33i  I  of  37  X  100  =  1233^ 

Multiply: 
24  by  331         27  by  33^         28  by  331- 
18  by  33i         21  by  33i         22  by  33^ 
30  by  33i         33  by  33^         35  by  33^ 

(e)  Find  the  sum  of  the  nine  products. 


PART   III.  403 

Art.  6.  To  multiply  a  number  by  34^ :     Take  33^  times 
the  member,  to  ivhich  add  the  numher  itself.     Why? 

24  X  341-       331  times  24  =.  800       800    +  24  =  824 

25  X  341       33-1-  times  25  =  833i     833i  +  25  =  8581 

Multiply : 
18  by  34-1-         27  by  34i-        24  by  341 
21  by  341-         33  by  34^         39  by  34^ 
30  by  341       ^  36  by  34i         15  by  34i 
12  by  34|         16  by  341         17  by  34i 

(f)  Find  the  sum  of  the  twelve  products. 

Art.  7.  To  multiply  a  number  by  35i :     Take  331  times 
the  numher,  to  which  add  twice  the  member.     Why  ? 

24  X  351       33^  times  24  =  800       800    +  48  =  848 

25  X  351       331-  times  25  =  8331     8331-  +  50  =  8831 

Multiply : 
21  by  351         30  by  351         27  by  35-1- 
18  by  351         66  by  35-i         36  by  35^ 
33  by  351         39  by  351         3I  by  35^ 

(g)  Find  the  sum  of  the  nine  products. 

Art.  8.  To  multiply  a  number  by  32 1:     Take  33-1  times 
the  number,  from  which  subtract  the  number  itself.     Why? 

21  X  321       331  times  21  =  700       700    -  21  =  679 

22  X  32^       331  times  22  =  7331     7331  _  22  =  7111 

Multiply : 

15  by  321         21  by  32i-         18  by  321- 

24  by  321         30  by  32-1         27  by  32i 

33  by  321         39  by  32-1         36  by  32i 

(h)  Find  the  sum  of  the  nine  products. 


404  COMPLETE    ARITHMETIC. 


Art.  9.  To  multiply  a 

L  number  b} 

^  25  :    Multiply  ^  of  the 

number  by  100.     Why? 

48  X  25 

i  of  48  X 

100  =  1200 

49  x25 

i  of  49  X 
Multiply : 

100  =  1225 

36  by  25      * 

32  by  25 

33  by  25 

40  by  25 

28  by  25 

29  by  25 

24  by  25 

16  by  25 

19  by  25 

52  by  25 

48  by  25 

50  by  25 

(i)  Find  the  sum  of  the  twelve  products. 

Art.  10.   To  multiply  a  number  by  26:     Take  25  times 
the  number,  to  which  add  the  member  itself.     Wliy  ? 

36x26       25  times  36  =  900       900  +  36  =  936 
37x26       25  times  37  =  925       925  +  37  =  963 

Multiply: 

36  by  26         48  by  26         45  by  26 

28  by  26         24  by  26         25  by  26 

44  by  26         32  by  26         35  by  26 

(j)  Find  the  sum  of  the  nine  products. 

Art.  11.   To  multiply  a  number  by  27:     Take  25  times 
the  number,  to  which  add  twice  the  member.     Why  ? 

36  X  27       25  times  36  =  900       900  +  72  =  972 

37  X  27 


27       25  times  37  =  925 

925  +  74:= 

=  999 

Multiply : 

48  by  27 

52  by  27 

37  by  27 

32  by  27 

16  by  27 

17  by  27 

28  by  27 

24  by  27 

26  by  27 

(k)  Find  the  sum  of  the  nine  products. 


PART   III.  405 

Art.  12.   To  multiply  a  number  by  24:     Take  25  times 
the  numhcr,  from  which  subtract  the  member  itself.     Why  ? 

32  X  24       25  times  32  =  800       800  -  32  :=  768 

33  X  24       25  times  33  =  825       825  -  33  =  792 


Multiply : 

24  by  24 

16  by  24 

17  by  24 

44  by  24 

36  by  24 

37  by  24 

28  by  24 

32  by  24 

35  by  24 

48  by  24 

52  by  24 

54  by  24 

(1)  Find  the  sum  of  the  twelve  products. 

Art.  13.  To  multiply  a  number  by  20  :  Multiply  i  of  the 
number  by  100.  How  may  a  number  be  multiplied  by  21? 
By  22?     By  19? 

35x21       20  times  35  =  700       700  +  35  =  735 

Multiply: 

45  by  21         45  by  22         45  by  19 

35  by  21         35  by  22         35  by  19 

36  by  21  36  by  22  36  by  19 
(m)  Find  the  sum  of  the  nine  products. 

Art.  14.  To  multiply  a  number  by  1 6| :  Multiply  J  of 
the  number  by  100.  How  may  a  number  be  multiplied  by 
17f?     BylSf?     By  15-1? 

24  X  17|       16f  times  24  =  400       400  +  24  =  424 

Multiply : 

18  by  17f         18  by  18f         18  by  15f 

30  by  17f         30  by  18f         30  by  15f 

36  by  17|  36  by  18|         36  by  15f 

(n)  Find  the  sum  of  the  nine  products. 


13^?     By  14^?     By 

11^? 

32  X  13i       12|- 

times  32  =  400 

Multiply : 

24  by  131 

24  by  141- 

16  by  13|- 

16  by  14^ 

40  by  13|- 

40  by  14|^ 

406  COMPLETE   ARITHMETIC. 

Art.  15.  To  multiply  a  number  by  12 J:  Multiply  i-  of 
the  numher  hy  100.     How  may  a  number  be  multiplied  by 

400  +  32  r:.  432 

24  by  11| 
16  by  14 
40  by  llj 
(0)  Find  the  sum  of  the  nine  products. 

Art.  16.  To  multiply  a  number  by  125  :  Multiply  ^  of 
the  numher  hy  1000.  How  may  a  number  be  multiplied  by 
126?     By  127?     By  124? 

96  X  125       -1-  of  96  X  1000  =  12000 

96  X  126     125  times  96  =  12000     12000  +  96  =  12096 

Multiply : 

120  by  126         120  by  127         120  by  124 

320  by  126         320  by  127         320  by  124 

240  by  126         240  by  127         240  by  124 

(p)  Find  the  sum  of  the  nine  products. 

Art.  17.  To  multiply  a  number  by  250  :  Multiply  ^  of 
the  numher  hy  1000.  How  may  a  number  be  multiplied  by 
251?     By  252?     By  249  ? 

48  X  250     i  of  48  x  1000  =  12000 

48  X  251     250  times  48  =  12000     12000  +  48  =  12048 

Multiply : 
60  by  251         60  by  252         60  by  249 
72  by  251  72  by  252         72  by  249 

(q)  Find  the  sum  of  the  six  products. 


PART   III.  407 

Art.  18.  To  square  2^,  S^,  4^,  etc.:   Multiply  the  integer 
by  the  integer  phis  1,  and  add  \  to  the  product. 


21-  X  21-  =  2  times  2+2  times  |-  +  ^of2+|-of^ 


But  2  times  -i-  +  ^  of  2  =  1  time  2  ;  and  ^  of  ^  =  \ 
Hence,  2|-  x  2^  =  2^^  +  t  =  ^i 
HxH  =  31^4  +  i  =  12^ 


Multiply : 

H  by  4^ 

H  by  5i 

6i  by  6J 

n  by  7i 

8t  by  8J 

9J  by  9^ 

H  by  H 

21-  by  2J 

3^  by  3^ 

(r)  Find  the  sum  of  the  nine  products. 

Art.  19.  To  square  25,  35,  45,  etc.:  Multiply  the  tens' 
figure^  hy  the  tens'  figure  increased  hy  1 ;  regard  the  product 
as  hundreds,  to  which  add  25. 

To  explain  this  rule,  think  of  25  as  2  tens  and  ^  of  a  ten, 
and  apply  the  explanation  given  under  Art.  18. 

25x25  =  2x3  hundred  and  25  =    625 


35  X  35  =  3  X  4  hundred  and  25  =  1225 


45  x  45  ==  4  X  5  hundred  and  25  =  2025 

Multiply : 

55  by  55         65  by  65         75  by  75 
85  by  85  95  by  95  15  by  15 

(s)  Find  the  sum  of  the  six  products. 

*The  author  is  aware  that  the  expressions  "Multiply  the  tens^  figure'^  and  "the 
tens'  figure  increased  by  1"  are  tabooed  by  the  hyi>ercritical.  But  it  is  believed  that 
neither  obscurity  nor  misconception  will  arise  from  this  use  of  the  word  figure.  The 
word  as  here  used  clearly  means  the  form  value  of  the  figure— the  number  which  the 
figure  hy  virtue  of  its  shape  represent?. 


408  COMPLETE    ARITHMETIC. 

Art.  20.  To  multiply  2|  by  2f ,  3  J-  by  3|-,  etc. :  Multiply 
tJi/i  integer  hy  the  integer  jplus  1,  and  to  the  product  add  the 
product  of  the  fractions. 

Observe  that  this  rule  will  apply  only  when  the  integer  of  the 
multiplicand  and  the  integer  of  the  multiplier  are  the  same,  and  the 
sum  of  the  fractions  is  1. 


H  by  2J  =  2  times  2  +  2  times  ^  +  |of2+|of| 
But  2  times  \  +  f  of  2  =  1  time  2,  and  f  of  |  =  f^ 
Hence,  2|  x  2f  =  2x3  +  f  of  i-  =  ^^% 


3|  X  3f  =  3  X  4  +  I  of  i  =  12/^ 
Multiply : 

^hJ^     ^hJ^     ^\^J^ 

7iby7f         8fby8i         H  by  9f 
(t)  Find  the  sum  of  the  six  products. 

Art.  21.  To  multiply  24  by  26.  33  by  37,  etc.:  Multiply 
the  tens'  figure  hy  the  tens'  figure  increased  hy  1 ;  regard  the 
product  as  hundreds,  to  which  add  the  product  of  the  units' 
figures. 

Observe  that  this  rule  will  apply  only  when  the  tens'  figure  of  the 
multiplicand  and  the  tens'  figure  of  the  multiplier  are  alike,  and  the 
sum  of  the  units'  figures  is  10. 


22  X  28  =  2  X  3  hundred  and  16  =    616 


33  X  37  =  3  X  4  hundred  and  21  =.  1221 

Multiply : 

21  by  29         23  by  27         24  by  26 

31  by  39         32  by  38         34  by  36 

41  by  49         42  by  48         43  by  47 

(u)  Find  the  sum  of  the  nine  products. 


PART  III.  409 

Art.  22.    To   multiply  a  number  by   15:    Multiply   the 
number  hy  1 0,  and  to  the  product  add  ^  of  the  product. 
64  X  15        10  times  64  =  640       640  +  320  =  960 
45x15       10  times  45  ^  450       450  +  225  =  675 

Multiply : 
24  by  15         32  by  15  35  by  15 

46  by  15         34  by  15         43  by  15 
82  by  15         66  by  15         75  by  15 
37  by  15         41  by  15         39  by  15 
(v)  Find  the  sum  of  the  twelve  products. 

Art.  23.  To  multiply  a  number  by  99:  Take  100  times 
the  number,  from  which  subtract  -the  number  itself.  How 
may  a  number  be  multiplied  by  98  ? 

36  X  99       100  times  36  =  3600       3600  -  36  =  3564 
42  X  98       100  times  42  =  4200       4200  -  84  =  4116 

Multiply : 
35  by  99         44  by  99         35  by  98 
'    27  by  99         54  by  99         46  by  98 
62  by  99  75  by  99  28  by  98 

(w)  Find  the  sum  of  the  nine  products. 

Art.  24.  To  multiply  a  number  by  75  :  Multiply  ^  of  the 
number  by  100.    How  may  a  number  be  multiplied  by  66-|? 


By  62|-? 

By87i? 

Multiply : 

64  by  75' 

24  by  66| 

64  by  874- 

48  by  75 

36  by  66| 

48  by  871- 

52  by  75 

63  by  66f 

56  by  87|- 

37  by  75 

37  by  66| 

32  by  87| 

(x)  Find  the  sum  of  the  twelve  products. 


410  COMPLETE    ARITHMETIC. 

Art.  25.  To  divide  a  number  by  25;  by  33i;  by  12^; 
by  16| ;  by  20 ;  by  50.  ^  (See  pp.  212,  213,  and  214,  of  this 
book.) 


850    - 

-25 
-331 

-m 

-16f 
-20 

-50 

=  8  times  4-^2 

=  34 

933i- 

=  9  times  3  -f-  1 

=  28 

637I-- 

=  6  times  8  -f  3 

=  51 

750    - 

=  7  times  6-^3 

=  45 

960    - 

=  9  times  5  +  3 

=  48 

450    - 

=  4  times  2  +  1 

=    9 

Divide : 
1275  by  25  1166|-  by  33i-  762|-  by  12i- 

950  by  16f  880    by  20  950    by  50 

(y)  Find  the  sum  of  the  six  quotients. 

Note. — Without  a  pencil,  tell  the  integral  quotient  and  the  re- 
mainder resulting  from  the  incomplete  division  of  1584  by  25. 

Art.  26.  To  divide  a  number  by  125;  by  250:  Observe 
that  125  is  contained  in  each  thousand  of  a  nnmher,  8  times; 
that  250  is  contained  in  each  thousand  of  a  number,  4  times. 


7125  -^  125  =  7  times  8  +  1  =  57 


8500  ^  250  =  8  times  4  +  2  =  34 

Divide : 

13250  by  250         13500  by  250         13750  by  250 

18000  by  250         18750  by  250         18500  by  250 

12000  by  125         12500  by  125         12625  by  125 

9125  by  125  9375  by  125  9875  by  125 

(z)  Find  the  sum  of  the  twelve  quotients. 

Note. — Without  a  pencil,  tell  the  integral  quotient  and  the  re- 
mainder resulting  from  the  incomplete  division  of  15450  by  250. 


PART   HI.  411 

Art.  27.  When  the  same  factor  occurs  in  a  dividend  and 
in  its  divisor,  it  may  be  omitted  from  both  without  changing 
their  ratio.  Hence  all  the  factors  that  are  common  to  a  div- 
idend and  its  divisor  may  he  stricken  out  (canceled)  and  the 
quotient  (ratio)  he  unchanged. 

Divide  180  by  42. 
Operation  No.  1.  Operation  No.  2. 

42)180(4-1  180  ^  ^X2x3x3x5  ^  30  ^  ^  2 

1^  42  ^X3x7  7         7 

12      2 

=r  —  Observe   that   the    striking   out    of   the 

4-^       '  factors  2  and  3  from  the  dividend  and  its 

divisor  does  not  change  their  ratio — the  quotient. 

II.    Divide  420  by  35. 
Operation  No.  1.  Operation  No.  2. 

35)420(12  420  ^  2x2x3x0x;?^  =  —  =  12 

35  35  ^xli  1 


70 

70 


Observe  that  if  all  the  factors  of  one  of 
the  numbers  are  canceled,  the  number  be- 
comes 1  and  not  0.  The  factor  5  is  5  times  1 :  the  factor  7,  7  times 
1.  Hence  in  the  above  problem  there  really  remain  in  the  divisor, 
after  the  cancellation,  the  factors  1  and  1  =  1x1=1- 

III.   Divide  48  x  8  x  4  =  1536  by  8  x  4  x  4  =  128. 
Operation  No.  1.  Operation  No.  2. 

12S);g6(12  ^xixi_12_^, 

$  X  ^  X  4  1 
Observe  that  it  is  not  necessary  to 
obtain  the  prime  factors  of  a  dividend 
and  its  divisor  to  employ  cancellation  in  finding  the  quotient.  In 
the  above  the  composite  factor  8  is  stricken  out  of  the  divisor  and 
out  of  the  48  of  the  dividend. 


256 
256 


412  COMPLETE    ARITHMETIC. 

IV.   Divide  56  x  35  =  1960  by  15  x  8  =  120. 
Operation  No.  1.  Operation  No.  2. 

120)1960(16.  M_xi^_49_^^l 

Ux^       3  3 


760 

'20  Observe  that  in  the  above  the  factor  5 

4Q        \  is  stricken  out  of  15   and  35,  and  the 

factor  8  is  stricken  out  of  the  divisor 
and  out  of  the  56  of  the  dividend. 


120       3 


Miscellaneous  Problems. 

Note. — Employ  "  Short  Methods  "  in  the  solution  of  the  follow- 
ing problems. 

How  many  cords  of  wood — 

1.  In  a  pile  32  feet  by  8  feet  by  4  feet  ?  * 

2.  In  a  pile  40  feet  by  1 6  feet  by  6  feet  ? 

3.  In  a  pile  32  feet  by  30  feet  by  10  feet  ? 
(aa)  Find  the  sum  of  the  three  results. 

How  many  acres  of  land — 

4.  In  a  piece  180  rods  by  28  rods?f 

5.  In  a  piece  64  rods  by  96  rods? 

6.  In  a  piece  136  rods  by  32  rods? 
(bb)  Find  the  sum  of  the  three  results. 

7.  Multiply  64  by  96  and  divide  the  product  by  16  x 
24  X  2. 

8.  Multiply  250  by  72  and  divide  the  product  by  16 1  x 
3  X  24. 

(cc)  Find  the  sum  of  the  two  results. 

*  Think  of  a  cord  as  8  feet  by  4  feet  by  4  feet. 
f  Think  of  an  acre  as  40  rods  by  4  rods. 


PART    III.  413 

Find  the  cost — 

9.  Of  346  acres  of  land  at  $50  per  acre. 

10.  Of  346  acres  of  land  at  $51  per  acre. 

11.  Of  346  acres  of  land  at  $52  per  acre. 

12.  Of  346  acres  of  land  at  $49  per  acre. 

13.  Of  254  acres  of  land  at  $51  per  acre, 
(dd)  Find  the  sum  of  the  five  results. 

14.  Of  243  ft.  iron  pipe  at  33-i^  a  foot. 

15.  Of  243  ft.  iron  pipe  at  3414  a  foot. 

16.  Of  243  ft.  iron  pipe  at  35^^  a  foot. 

17.  Of  243  ft.  iron  pipe  at  32^^  a  foot. 

18.  Of  156  ft.  iron  pipe  at  35^^  a  foot, 
(ee)  Find  the  sum  of  the  five  results. 

19.  Of  260  lb.  butter  at  25^  a  pound. 

20.  Of  260  lb.  butter  at  26^  a  pound. 

21.  Of  260  lb.  butter  at  27^  a  pound. 

22.  Of  260  lb.  butter  at  24^  a  pound. 

23.  Of  184  lb.  butter  at  27^-  a  pound, 
(ff)  Find  the  sum  of  the  five  results. 

24.  Of  350  lb.  coffee  at  12 J^  a  pound. 

25.  Of  350  lb.  coffee  at  13|^^  a  pound. 

26.  Of  350  lb.  coffee  at  14J^  a  pound. 

27.  Of  350  lb.  coffee  at  11^^  a  pound. 

28.  Of  330  lb.  coffee  at  16|^  a  pound. 

29.  Of  330  lb.  coffee  at  17f^  a  pound. 

30.  Of  330  lb.  coffee  at  15f^  a  pound, 

31.  Of  240  lb.  coffee  at  25^  a  pound. 

32.  Of  240  lb.  coffee  at  26^  a  pound. 

33.  Of  240  lb.  coffee  at  27^  a  pound, 
(gg)  Find  the  sum  of  the  ten  results. 


414  COMPLETE    AKITHMETIC. 

Find  the  cost — 

34.  Of  2i  tons  coal  at  $2^  per  ton. 

35.  Of  3|-  tons  coal  at  $3|-  per  ton. 

36.  Of  4  J  tons  coal  at  $4^  per  ton. 
(hli)  Find  the  sum  of  the  three  results. 

37.  Of  25  tons  of  meal  at  $25  per  ton. 

38.  Of  35  acres  of  land  at  $35  per  acre. 

39.  Of  45  M.  ft.  of  lumber  at  $45  per  M 
(ii)  Find  the  sum  of  the  three  results. 

40.  Of  23  yd.  cloth  at  27^  a  yard. 

41.  Of  36  yd.  cloth  at  34^  a  yard. 

42.  Of  42  yd.  cloth  at  48^  a  yard, 
(jj)  Find  the  sum  of  the  three  results. 

43.  Of  3240  ft.  lumber  at  $15  per  M. 

44.  Of  2460  ft.  lumber  at  $15  per  M. 

45.  Of  1620  ft.  lumber  at  $16  per  M. 
(kk)  Find  the  sum  of  the  three  results. 

46.  Of  99  lb.  butter  at  23^  a  pound. 

47.  Of  99  lb.  butter  at  28^  a  pound. 

48.  Of  98  lb.  butter  at  24^  a  pound. 
(11)  Find  the  sum  of  the  three  results. 

49.  Paid  $15.50  for  ribbon  at  16|^  a  yard.  How  many 
yards  did  I  buy  ? 

60.  Paid  $24.75  for  ribbon  at  12|^  a  yard.  How  many 
yards  did  I  buy  ? 

(mm)  Find  the  sum  of  the  two  results. 


PRACTICAL  APPROXIMATIONS. 

So  far  as  practicable,  solve  the  following  problems  without  the 
aid  of  a  pencil.  At  least,  exercise  the  judgment  on  every  problem 
before  making  any  figures^. 

1.  The  specific  gravity*  of  iron  being  about  1\,  how  much 
does  a  cubic  foot  of  it  weigh  ?  How  much  does  a  cubic 
inch  of  iron  weigh  ? 

2.  A  4-inch  iron  ball  weighs  about  pounds.     A  2- 

inch  iron  hall  weighs  about  pounds. 

3.  An  iron  rod,  1  inch  in  diameter  and  12  feet  long, 
weighs  about  pounds.  An  iron  rod  2  inches  in  diam- 
eter and  12  feet  long  weighs  about  pounds. 

4.  A  sheet  of  boiler  iron,  8  feet  square  and  f  of  an  inch 
thick,  weighs  about  pounds. 

5.  What  is  the  weight  of  the  water  that  will  fill  a  tank 
2  feet  wide,  2  feet  deep,  and  1 0  feet  long  ? 

6.  The  specific  gravity  of  limestone  is  about  2^.  What 
is  the  weight  of  a  piece  of  limestone  that  is  4  feet  square 
and  3  inches  thick  ? 

7.  The  specific  gravity  of  seasoned  white  pine  is  about  .5 ; 
that  is,  a  piece  of  white  pine  weighs  about  5  tenths  as  much 
as  the  same  bulk  of  water  weighs.  How  much  does  a  pine 
board  1  foot  wide,  1  inch  thick,  and  12  feet  long,  weigh? 

8.  What  is  the  weight  of  a  stick  of  timber  12  inches  by 
12  inches  and  20  feet  long,  if  its  specific  gravity  is  .7  ? 

9.  What  is  the  weight  of  1000  feet  of  green  lumber  if  its 
specific  gravity  is  .9  ? 

*When  we  say  that  the  specific  gravity  of  iron  is  about  1\,  we  mean  that  it 
weighs  about  71  times  as  much  as  water,  the  same  bulk  being  considered. 

415 


416  COMPLETE    ARITHMETIC. 

10.  The  specific  gravity  of  sand  is  about  2.  How  much 
does  a  load  (27  cu.  ft.)  of  it  weigh  ? 

11.  If  the  specific  gravity  of  granite  is  2.7,  how  much 
does  a  cubic  yard  of  it  weigh  ? 

12.  The  specific  gravity  of  brick  and  mortar  is  nearly  2. 
What  is  the  weight  of  a  cubic  foot  of  brick  wall  ? 

13.  If  a  brick,  8  in.  by  4  in.  by  2  in.,  weighs  4J  pounds, 
is  its  specific  gravity  more  or  less  than  2  ?  That  is,  does  1 
cubic  foot  of  bricks  weigh  more  or  less  than  exactly  twice 
as  much  as  1  cubic  foot  of  water  ? 

14.  If  the  specific  gravity  of  Athens  (Illinois)  limestone 
is  2.4,  and  if  a  cord  of  it  is  equal  to  100  soM  feet,  how  much 
does  a  cord  of  this  stone  weigh  ? 

15.  If  bricks,  8  in.  long,  4  in.  wide,  and  2  in.  thick,  are 
laid  on  their  largest  face,  how  many  bricks  will  be  required 
for  a  walk  6  feet  wide  and  100  feet  long,  making  some 
allowance  for  imperfect  bricks  and  breakage  in  handling? 

16.  What  is  the  capacity  in  gallons  of  a  tank  5  feet  long, 
2  feet  wide,  and  2  feet  deep  ? 

17.  Give  the  dimensions  of  a  tank  the  capacity  of  which 
is  225  gallons. 

18.  A  certain  cylindrical  tank  is  8  feet  in  diameter. 
Each  foot  in  depth  will  contain  how  many  gallons  ? 

19.  One  inch  of  rain-fall  will  give  how  many  pounds  of 
water  on  a  horizontal  surface  30  feet  by  40  feet  ? 

20.  Two  inches  of  rain-fall  will  give  how  many  barrels 
(31 1^  gal.)  of  water  on  a  horizontal  surface  40  feet  by  60 
feet? 

21.  A  one  inch  rain-fall  will  give  how  many  tons  of 
water  to  the  acre  ? 

22.  Your  school-house  lot  is  what  part  of  an  acre  ? 


PART    III.  417 

.23.  A  one   inch   rain-fall  will   give   how   many    barrels 
(31|-  gal.)  of  water  on  the  school-house  lot  ? 

24.  Your  school-room  floor  is  what  part  of  an  acre  ? 

25.  The  distance  around  your  school-room  is  what  part  of 
a  mile  ? 

26.  How  many  bushels  of  oats  would  be  required  to  till 
your  school-room  to  the  depth  of  3  feet  ? 

27.  A  shed  as  large  as  your  school-room  would  hold  how 
many  cords  of  wood  ? 

28.  Give  the  dimensions  of  a  crib  that  will  hold  1000 
bushels  of  corn. 

29.  Give  the  dimensions  of  a  pile  of  wood  that  contains 
40  cords. 

30.  Apiece  of  land  100  feet  long  and  5  rods  wide  is  what 
part  of  an  acre  ? 

31.  How  many  cubic  inches  in  a  cylinder  8  inches  in 
diameter  and  10  inches  long? 

32.  How  many  cubic  inches  in  an  8-inch  sphere  ? 

33.  If  468.25  be  multiplied  by  .5106  will  the  product  be 
more  or  less  than  234  ?  * 

34.  If  484,079  be  multiplied  by  .251,  will  the  product 
be  more  or  less  than  121  ? 

35.  If  2480  be  multiplied  by  .2479  will  the  product  be 
more  or  less  than  620  ? 

36.  If  6400  be  multiplied  by  .74,  the  product  will  be 
how  many  less  than  ^  of  6400  ? 

37.  If  4800  be  multiplied  by  1.6,  the  product  will  be 
how  many  more  than  1|-  times  4800  ? 

38.  If  366.06  be  divided  by  |,  will  the  quotient  be  more 
or  less  than  366  +  J-  of  366  ? 

*  Observe  that  234  is  J  of  468. 


418  COMPLETE    ARITHMETIC. 

39.  If  25.2314  be  divided  by  J,  will  the  quotient  be  more 
or  less  than  51  ? 

40.  If  250  be  divided  by  .26,  will  the  quotient  be  more 
or  less  than  four  times  250  ? 

41.  If  cheese  is  worth  17  cents  a  pound,  how  much 
should  be  paid  for  (a)  2  lb.  3  oz.?  (b)  3  lb.  5  oz.?  (c)  1  lb. 
7  oz.?     (d)  4  lb.  9  oz.     (e)  2  lb.  11  oz.? 

42.  If  meat  is  worth  15  cents  a  pound,  how  much  should 
be  paid  for  (a)  1  lb.  4  oz.?  (b)  2  lb.  6  oz.?  (c)  1  lb.  7  oz.? 
(d)  2  lb.  9  oz.?     (e)  3  lb.  13  oz.? 

43.  If  cheese  is  worth  12|^  cents  a  pound,  how  much 
should  be  paid  for  (a)  2  lb.  8  oz.?  (b)  2  lb.  4  oz.?  (c)  3  lb. 
1  oz.?     (d)  4  lb.  9  oz.?     (e)  4  lb.  15  oz.? 

44.  A  wagon  box  10  feet  by  3  feet  by  16  inches  will 
contain  how  many  bushels  of  shelled  corn? 

45.  The  surface  of  a  sphere  is  equal  to  4  times  the  area 
of  a  circle  having  the  same  diameter  as  the  sphere.  How 
many  square  inches  in  the  surface  of  a  10-inch  sphere? 

46.  How  many  square  inches  in  the  surface  of  a  20-inch 
sphere  ? 

47.  If  from  a  cylinder  of  wood  the  largest  possible  cone 
be  cut,  exactly  -|  of  the  wood  will  be  cut  away.  The  solid 
content  of  a  cone  is  therefore  exactly  ^  of  a  cylinder  having 
the  same  base  and  the  same  altitude  (length).  How  many 
cubic  inches  in  a  cone  the  diameter  of  whose  base  is  8 
inches  and  whose  altitude  is  12  inches? 

48.  How  many  bushels  of  grain  in  a  conical  pile  whose 
diameter  is  6  feet  and  whose  altitude  is  4  feet  ? 

49.  At  10  cents  a  square  yard,  what  is  the  cost  of  paint- 
ing the  outer  surface  of  a  cylindrical  standpipe  whose 
diameter  is  15  feet  and  whose  altitude  is  36  feet? 


PAET   III.  419 

50.  At  1 0  cents  a  square  foot,  what  is  the  cost  of  lining  a 
cylindrical  tank  (curved  surface  and  bottom)  whose  diameter 
is  10  feet  and  whose  depth  is  12  feet? 

51.  A  sphere  is  exactly  ^  and  a  cone  exactly  -i-  of  a  cyl- 
inder of  the  same  dimensions.  Find  the  solid  content  and 
compare  the  following : 

(a)  A  6 -inch  cube. 

(b)  A  cylinder  6  in.  in  diameter  and  6  in.  long. 

(c)  A  6 -inch  sphere. 

(d)  A  cone ;  base  6  in.  in  diameter,  altitude  6  in. 

52.  A  cubic  foot  of  steel  weighs  490  lb.  (a)  What  is  the 
weight  of  a  cylinder  of  steel  1  foot  in  diameter  and  1  foot 
long  ?  (b)  Of  a  sphere  of  steel  1  foot  in  diameter  ?  (c)  Of 
a  cone  of  steel,  base  1  foot  in  diameter,  altitude  1  foot  ? 

53.  A  circular  piece  of  land  20  rods  in  diameter  contains 
more  or  less  than  2  acres  ? 

54.  At  sight,  give  approximate  answers  to  the  following : 

(a)  Interest  of  $450.25  for  1  yr.  3  da.  at  6%  ? 

(b)  Interest  of  $4000  for  29  da.  at  6%  ? 

(c)  Interest  of  $250  for  1  yr.  8  mo.  5  da.  at  6%  ? 

(d)  Interest  of  $500  for  1  yr.  11  mo.  16  da.  at  6%  ? 

(e)  Interest  of  $751.27  for  2  yr.  6  mo.  1  da.  at  4%  ? 

(f)  Interest  of  $149.75  for  1  yr.  7  mo.  29  da.  at  6%  ? 

(g)  Interest  of  $298.97  for  1  yr.  6  mo.  4  da.  at  8%  ? 
(h)  Interest  of  $495  for  15  da.  at  6%  ? 

(i)    Interest  of  $1200  for  6  da.  at  8%  ? 

(j)    Interest  of  $600  for  20  da.  at  4%  ? 

(k)  Interest  of  $397.28  for  2  yr.  6  mo.  at  6%  ? 

(1)    Interest  of  $5000  for  5  mo.  29  da.  at  8%  ? 

(m)  Interest  of  $3000  for  2  mo.  29  da.  at  7%  ? 

(n)  Interest  of  $4000  for  3  mo.  29  da.  at  6%  ? 


420  COMPLETE    ARITHMETIC. 

55.  At  sight,  give  answers  to  the  following,  that  are  true 
to  dollars;  then,  with  the  aid  of  a  pencil,  if  necessary,  obtain 
answers  that  are  true  to  cents. 

(a)  Cost  of  2970  lb.  coal  at  $4.50  per  ton  V 

(b)  Cost  of  3520  lb.  hay  at  $11.50  per  ton?'-^ 

(c)  Cost  of  1490  lb.  straw  at  $4.25  per  ton  ? 

(d)  Cost  of  2460  lb.  bran  at  $10.00  per  ton?^ 

(e)  Cost  of  2310  lb.  oil  meal  at  $21  per  ton  ? 

(f)  Cost  of  2240  lb.  beef  at  $6.10  per  cwt.? 
-     (g)  Cost  of  1560  lb.  pork  at  $4.50  per  cwt.? 

(h)  Cost  of  2150  lb.  flour  at  $3.05  per  cwt.? 
(i)    Cost  of  1200  lb.  lard  at  $5.90  per  cwt.? 
(j)    Cost  of  1400  lb.  tallow  at  $3.55  per  cwt.? 
(k)  Cost  of  1000  lb.  nails  at  $3.05  per  cwt.? 
(1)    Cost  of  2240  ft.  lumber  at  $15  per  M.?' 
(m)  Cost  of  4500  lath  at  $2.50  per  M.? 
(n)  Cost  of  6740  brick  at  $6.00  per  M.?' 
(o)  Cost  of  1997  ft.  lumber  at  $27.50  per  M.? 
(p)  Cost  of  198  lb.  butter  at  27|-  cts.  per  lb.?' 
(q)  Cost  of  20  3|-  lb.  cheese  at  16  cents  per  lb.? 
(r)  Cost  of  2440  lb.  oats  at  24 1-  cents  per  bu.?' 
(s)   Cost  of  1680  lb.  oats  at  23^  cents  per  bu.? 

>  2970  lb.  is  nearly  IJ  tons. 

2  What  is  the  cost  of  3500  at  $12  per  ton  ? 

3  At  $10  a  ton,  how  much  does  1  lb.  cost? 

*  At  $15  i)er  M.,  how  much  is  1  foot  worth? 

"  6740  brick  are  nearly  62  M. 

«  200  lb.  butter  at  27J  cents  is  worth  how  much  ? 

'  At  24^  a  bushel,  1  lb.  of  oats  is  worth  how  much  ? 


MISCELLANEOUS    PEOBLEMS. 

Note. — The  following  problems  are  selected  mainly  from  sets  of 
examination  questions  supplied  to  the  author  for  this  purpose  by 
one  hundred  school  principals  and  superintendents. 

1.  Each  edge  of  a  cube  is  diminished  by  -^^  of  its  length. 

(a)  By  what  fraction  of  itself  is  the  volume  diminished  ? 

(b)  By  what  fraction  of  itself  is  the  surface  diminished  ? 

2.  How  many  cubical  blocks,  each  edge  of  which  is  ^  ft., 
are  equivalent  to  a  block  8  ft.  long,  4  ft.  wide,  and  2  ft. 
thick  ? 

3.  A  ladder  78  ft.  long  stands  perpendicularly  against  a 
building.  How  far  must  it  be  pulled  out  at  the  foot  that  the 
top  may  be  lowered  6  ft.  ? 

4.  A  merchant  sold  |  of  a  quantity  of  cloth  at  a  gain  of 
20%  and  the  remainder  at  cost. 

(a)  His  gain  was  what  per  cent  of  the  cost  ? 

(b)  If  he  gained  $7.29  what  was  the  cost  of  the  goods  ? 

5.  What  must  I  pay  for  4%  stock  to  get  5%  on  the 
investment  ? 

6.  The  cubical  content  of  one  cube  is  eight  times  that  of 
another : 

(a)  How  does  an  edge  of  the  first  compare  with  an  edge  of 
the  second  ? 

(b)How  does  the  surface  of  the  first  compare  with  the 
surface  of  the  second  ? 

7.  A  creditor  receives  $1.50  for  every  $4.00  that  is  due 
him  and  thereby  loses  $301.05. 

(a)  What  was  the  sum  due  him?  (b)  What  per  cent  of 
the  debt  did  he  lose  ? 

421 


422  COMPLETE    ARITHMETIC. 

8.  At  $20  per  M.,  board  measure,  what  is  the  cost  of  the 
following:  A  stick  of  timber  30  feet  long  and  14  inches 
square,  and  a  plank  18  feet  long,  8  inches  wide,  and  2^ 
inches  thick  ? 

9.  A  and  B  hire  a  pasture  for  $85 ;  A  puts  in  -8  cows 
and  B  puts  in  12  cows.     How  much  should  each  pay  ? 

10.  Simplify  the  following :  ^  of  J-  of  li. 
•  11.  Seven  times  John's  property  plus  $32200  equals  21 
times  his  property.     How  much  is  he  worth  ? 

12.  Two  men  engage  in  business  with  a  joint  capital  of 
$5000.  The  first  year's  gain  was  $1760,  of  which  one 
received  $1056.     How  much  capital  did  each  furnish? 

13.  Thirty-five  per  cent  of  the  men  in  a  regiment  being 
sick,  only  637  men  were  able  to  enter  battle.  How  many 
men  were  there  in  the  regiment  ? 

14.  A  lawyer  collected  80%  of  a  debt  of  $2360  and 
charged  5%  commission  on  the  sum  collected.  How  much 
did  the  creditor  receive  ? 

15.  Write  a  negotiable  note  for  $500,  making  yourself  the 
payee  and  James  J.  Kogers  the  maker.  Interest  at  the  legal 
rate. 

16.  A  speculator  bought  stock  at  25%  below  par  and  sold 
it  at  20%  above  par.  He  gained  $1035.  How  much  did 
he  invest  ? 

17.  What  is  the  rate  per  cent  per  annum  if  $712  gains 
$142.40  in  3  yr.  4  mo.? 

18.  A  person  asked  for  a  lot  of  land,  40  %  more  than  it  cost 
him,  but  finally  reduced  his  price  15%  of  his  asking  price 
and  sold  it,  making  $9.50. 

(a)  What  per  cent  did  he  make  ?  (b)  How  much  did  the 
land  cost  him  ?     (c)  How  much  did  he  receive  for  it  ? 


PART   III.  423 

19.  Purchased  stock  at  a  premium  of  8  per  cent.  What 
rate  of  interest  do  I  receive  on  the  investment  if  it  pays  an 
annual  dividend  of  6  %  ? 

20.  Find  the  vokime  of  a  cube  the  area  of  whose  surface 
is  100.86  square  inches. 

21.  How  many  apples  must  a  boy  buy  and  sell  to  make  a 
profit  of  S9.30,  if  he  buys  at  the  rate  of  5  for  3^  and  sells  at 
the  rate  of  4  for  3^  ? 

22.  Find  the  cost  of  1875  lb.  hay  at  $6.50  per  ton. 

23.  What  is  the  interest  on  $1200  from  Sept.  21,  1898,  to 
May  5,  1899,  at  7%  per  annum  ? 

24.  The  area  of  a  square  field  is  10  acres.  What  is  the 
distance  diagonally  across  the  field  ? 

25.  A  "drummer"  earns  $2500  a  year.  One  thousand 
dollars  of  this  sum  is  a  guaranteed  salary.  The  remainder 
is  his  commission  of  5%  on  his  sales.  What  is  the  amount 
of  his  annual  sales  ? 

26.  The  area  of  a  triangle  is  325  square  inches.  Its  base 
is  25  inches.     What  is  its  altitude  ? 

27.  Gave  6|-  lb.  butter,  worth  36^'  a  pound,  for  3^  gal.  oil. 
What  was  the  cost  of  the  oil  per  gallon  ? 

28.  A  can  build  a  certain  wall  in  10  days  ;  B  can  build  it 
in  12  days,  and  C  in  15  days.  In  how  many  days  can  they 
build  the  wall  working  together  ? 

29.  A  horse  and  a  carriage  together  cost  $550.  The  horse 
cost  I  as  much  as  the  carriage.     Find  the  cost  of  each  ? 

30.  A  man  willed  J  of  his  property  to  his  wife,  ^  of  the 
remainder  to  his  daughter,  and  the  rest  to  his  son.  The 
difference  between  the  wife's  portion  and  the  son's  portion 
was  $12480.331     How  much  was  the  man  worth? 


424  COMPLETE    AKITHMETIC. 

31.  (a)  How  many  loads,  each  containing  a  cubic  yard, 
will  be  required  to  fill  a  street  150  feet  long,  50  feet  wide, 
and  2^  feet  deep?  (b)  How  much  will  it  cost  at  18^  per 
cubic  yard  ? 

32.  A  right  triangle  has  two  equal  sides.  Its  hypothenuse 
is  100  rods  long,  (a)  Find  one  of  its  two  equal  sides,  (b) 
Find  its  area. 

33.  What  is  the  ratio  of  the  area  of  a  circle  to  the  area  of 
its  circumscribed  square  ? 

34.  What  is  the  ratio  of  the  square  of  the  radius  to  the 
square  of  the  diameter  of  the  same  circle  ? 

35.  A  man's  tax  is  $37.50.  The  rate  of  tax  is  li%. 
Property  is  assessed  at  30%  of  its  value.  What  is  the 
man's  property  worth  ? 

36.  A  field  containing  160  acres  is  40  rods  wide.  At  45^ 
a  rod,  how  much  less  would  it  cost  to  fence  a  square  field 
containing  the  same  number  of  acres? 

37.  My  agent  in  Baltimore  having  sold  a  consignment  of 
grain,  after  taking  out  his  commission  at  3  %  and  paying  a 
freight  bill  of  $1,125.00,  sent  me  a  draft  for  the  amount  due 
me — $19,536.00.     For  how  much  was  the  grain  sold  ? 

38.  How  many  pickets  4  inches  wide,  placed  3  inches 
apart,  are  required  to  fence  a  garden  21  rods  long  and  14 
rods  wide? 

39.  At  $6.30  a  cord,  what  is  the  value  of  wood  that  can  be 
piled  under  a  shed  50  ft.  long,  25  ft.  wide,  and  12  ft.  high? 

40.  Find  the  curved  surface  of  a  cylinder  6  ft.  in  diameter 
and  12  ft.  long. 

41.  (a)  What  is  the  bank  discount  and  (b)  what  are  the 
proceeds  on  a  note  for  $125  payable  in  90  days,  the  rate  of 
discount  being  8  %  ? 


PART   III.  425 

42.  If  sugar  that  cost  5^  a  pound  is  sold  at  18  lb.  for  a 
dollar,  what  is  the  gain  per  cent  ? 

43.  What  is  the  area  of  a  circle  30  inches  in  diameter  ? 

44.  What  is  the  volume  of  a  12-inch  globe  ? 

45.  What  is  the  circumference  of  a  circle  that  is  40  rods 
in  diameter  ? 

46.  The  "  number  belonging  "  in  a  certain  school  was  74. 
Five  were  absent  in  the  forenoon  and  seven  in  the  afternoon. 
Wliat  was  the  per  cent  of  attendance  for  the  day  ? 

47.  A  rectangular  piece  of  land  41  rods  by  24  rods  is 
how  many  acres  ? 

48.  Find  the  value  of  x  in  the  following  proportion: 
17.5  :  25  ::^:  40. 

49.  A  circular  ^-mile  race  track  encloses  how  many  acres  ? 

50.  If  the  same  number  be  added  to  the  numerator  and 
to  the  denominator  of  a  proper  fraction,  will  it  make  the 
fraction  greater  or  less  ? 

51.  A  certain  roof  is  40  feet  long  and,  measured  horizon- 
tally, 24  feet  in  width.  A  2-inch  rain-fall  should  give  how 
many  inches  in  depth  in  a  cistern  that  receives  the  water 
from  this  roof,  the  cistern  being  6  feet  long  and  4  feet 
wide? 

62.  A  can  do  a  piece  of  work  in  i  of  a  day.  B  can  do 
the  same  amount  of  work  in  \  of  a  day.  In  what  part  of  a 
day  can  both  working  together  do  the  piece  of  work  ? 

53.  C  can  do  a  certain  piece  of  work  in  3  days.  D  can 
do  the  same  amount  of  work  in  4  days.  In  how  long  a 
time  can  both  working  together  do  the  piece  of  work  ? 

54.  If  the  6-foot  drive  wheel  of  a  locomotive  makes  840 
revolutions  in  moving  a  certain  distance,  how  many  revolu- 
tions will  a  7-foot  wheel  make  in  moving  the  same  distance? 


426  COMPLETE    ARITHMETIC. 

55.  The  circumference  of  one  of  my  carriage  wheels  is  12 
feet.  The  circumference  of  another  wheel  on  the  same 
carriage  is  14  feet.  How  far  has  the  carriage  run  when  the 
smaller  wheel  has  made  exactly  300  more  revolutions  than 
the  larger  wheel  ? 

56.  Is  the  capacity  of  a  cylindrical  pail  6  inches  in  diam- 
eter and  5  inches  deep  more  or  less  than  ^  of  a  gallon  ?  . 

57.  At  what  rate  per  cent  must  I  invest  $800  that  in  1 
year  6  months  it  will  amount  to  $854  ? 

58.  If  exactly  |  of  a  stick  of  timber  floating  in  the  water 
is  submerged,  and  if  the  timber  is  12  inches  by  12  inches 
and  30  feet  long,  how  many  pounds  does  it  weigh  ? 

Note. — If  f  of  the  timber  is  submerged,  it  weighs  f  as  much  as 
its  own  bulk  of  water. 

59.  How  many  pickets  are  required  to  inclose  a  square 
2|-acre  lot  if  the  pickets  are  3  inches  wide  and  3  inches 
apart  ? 

60.  The  boundary  of  a  certain  field  is  described  as  follows: 
Beginning  at  the  northeast  corner  of  section  14 ;  thence 
south,  24  rods;  thence  west,  20  rods;  thence  south,  15 
rods;  thence  west,  40  rods;  thence  south,  41  rods;  thence 
west,  20  rods;  thence  north,  80  rods;  thence  east,  80 
rods,  to  the  place  of  beginning.  How  many  acres  in  the 
field? 

61.  At  90  cents  a  yard,  find  the  cost  of  carpeting  a  room 
that  is  15  feet  wide  and  18  feet  long,  the  cajpet  to  run 
lengthwise  of  the  room,  there  being  a  w^aste  of  1  foot  on 
each  breadth,  except,  the  first,  for  matching;  carpet  1  yd.  wide. 

62.  The  perimeter  of  a  rectangular  field  is  144  rods  and 
its  length  is  twice  its  breadth.     Find  its  area. 


PART    III.  427 

63.  Change  f  of  a  mile  to  a  compound  number  made  up 
of  rods,  feet,  and  inches. 

64.  Sold  1^  of  a  barrel  of  sugar  for  what  ^  of  it  cost. 
What  was  the  per  cent  of  loss  ? 

65.  Sold  |-  of  a  barrel  of  sugar  for  what  |  of  it  cost. 
What  was  the  per  cent  of  gain  ? 

66.  If  a  merchant  sells  goods  at  a  uniform  profit  of  20%, 
and  his  sales  on  a  certain  day  amount  to  $60,  his  gain  is 
how  many  dollars  ? 

67.  Find  the  cost  of  600  ft.  of  gas  pipe,  list  28^  a  foot,  at 
"55  and  3  lO's  off."* 

68.  Bought  for  "  60  off  "  and  sold  for  "  50  off."  What 
was  the  per  cent  of  profit  ? 

69.  From  |^  of  a  certain  number  subtract  f  of  it  and  27 
remains.     What  is  the  number  ? 

70.  Divide  the  number  495  into  two  parts,  the  ratio  of 
the  parts  being  as  2  to  3. 

71.  Divide  the  number  187  into  two  parts,  the  ratio  of 
the  parts  being  as  -|  to  f . 

72.  The  product  of  a  certain  number  multiphed  by  If  is 
352.     What  is  the  number  ? 

73.  Find  the  cost  at  $16  per  M.  of  2-in.  plank  for  a  floor 
24  feet  by  42  feet. 

74.  The  perimeter  of  an  oblong  is  192  ft.,  and  its  length 
is  twice  its  breadth.     Find  its  area. 

75.  At  $45  per  M.,  find  the  cost  of  a  board  16  feet  long, 
18  inches  wide,  and  14-  inches  thick. 

76.  A  man  bought  a  horse  and  a  carriage  for  $315  ;  he 
paid  2i  times  as  much  for  the  carriage  as  for  the  horse. 
Find  the  cost  of  each. 

*  "  55  and  3  lO's  oflf,"  means  "  55  and  10  and  10  and  10  off.' 


428  COMPLETE    ARITHMETIC. 

77.  A  house  and  lot  cost  $5000.  For  how  much  per 
month  must  it  rent  to  pay  the  owner  a  sum  equal  to  5  %  of 
its  cost  and  $230  for  insurance,  taxes,  and  repairs? 

78.  If  4 J-  lb.  of  butter  can  be  made  from  100  lb.  of  milk, 
how  much  butter  per  week  can  be  made  in  a  creamery  that 
is  receiving  15000  lb.  of  milk  a  day? 

79.  The  area  of  a  rectangular  piece  of  land  36  rods  long 
is  900  square  rods.  How  many  rods  of  fence  required  to 
enclose  the  field? 

80.  A  shingle  is  4  inches  wide.^     If  shingles  are  laid  4|- 

inches  to  the  weather,  each  shingle  practically  covers  

square  inches.     Then  shingles  will  cover  1  square  foot. 

On  account  of  waste  and  short  measurements,  it  is  necessary 
to  purchase  9  shingles  for  every  square  foot  to  be  covered,  if 
they  are  to  be  laid  4^  inches  to  the  weather.  Shingles  are 
put  up  in  bunches  20  inches  wide  and  containing  50  courses. 
Hence  each  bunch  contains  250  shingles.  At  the  lumber 
yards  parts  of  bunches  are  not  offered  for  sale.  How  many 
bunches  of  shingles  must  I  purchase  for  a  double  roof  35 
feet  long,  rafters  16  feet  long,  the  shingles  to  be  laid  4^ 
inches  to  the  weather  ? 

81.  The  square  of  a  certain  number  is  576.  What  is  its 
cube? 

82.  The  area  of  one  face  of  a  cube  is  64  square  inches. 
What  is  the  solid  content  of  the  cube  ? 

83.  In  plowing  an  acre  with  a  twelve-inch  plow  the  man 
walking  behind  it  travels  (43560  -^  5280)  8^  miles.  What 
part  of  8^  miles  will  that  man  travel  who  plows  an  acre 
with  a  14-inch  plow?     With  a  16 -inch  plow? 

*  Shingles  are  not  of  uniform  width  ;  but  in  counting  them  at  the  lumber  yards, 
every  4  inches  in  width  is  called  1  shingle. 


PART  III.  429 

84.  A  can  do  a  piece  of  work  in  12  days.  A  and  B  can 
do  an  equal  amount  of  work  in  8  days.  In  how  long  a  time 
can  B  do  the  work? 

85.  The  edge  of  one  cube  is  2|^  times  as  long  as  the  edge 
of  another  cube,  (a)  The  surface  of  the  first  cube  is  how 
many  times  the  surface  of  the  second  cube  ?  (b)  The  solid 
content  of  the  first  cube  is  how  many  times  the  solid  con- 
tent of  the  second  cube? 

86.  At  "  50  and  10  off"  the  net  cost  was  $29.34.  Find 
the  list  price. 

87.  How  many  160-acre  farms  in  a  township  6  miles  long 
and  6  miles  wide  ? 

88.  If  there  is  a  4-rod  road  on  every  section  line*  of  a 
township  6  miles  square,  (a)  how  many  acres  of  the  town- 
ship in  its  roads  ?  (b)  How  many  acres  of  each  square  160- 
acre  farm  are  taken  for  roads? 

89.  What  single  discount  is  equivalent  to  "40  and  20 
and  10  off"? 

90.  Estimate  the  weight  of  a  4-inch  sod  from  an  acre  of 
ground. 

91.  A  room  16  feet  by  22  feet  has  a  floor  made  of  4-inch 
tile,  (a)  How  many  tiles  in  the  floor?  (b)  How  many  tiles 
in  the  border  of  four  rows  ? 

92.  If  I  buy  at  20%  below  list  price  and  sell  at  20% 
above  list  price,  what  is  my  per  cent  of  gain  ? 

93.  Add  two  hundred  and  seven  thousandths,  and  two 
hundred  seven  thousandths. 

94.  From  nine  hundred  and  eight  ten-thousandths,  sub- 
tract nine  hundred  eight  ten-thousandths. 

*  A  section  is  1  mile  square,  and  half  of  the  width  of  the  road  is  on  each  side  of 
every  section  line. 


430  COMPLETE   ARITHMETIC. 

95.  Add  two  hundred  seventy-five  tenths,  three  hundred 
twenty-four  hundredths,  and  five  hundred  thirty-six  thou- 
sandths. 

96.  Multiply  six  hundred  twenty-seven  and  forty-five 
thousandths  by  two  and  six  tenths. 

97.  Divide  one  hundred  forty-four  by  twelve  hundredths. 

98.  If  the  hills  of  corn  are  3|-  feet  apart  each  way,  (a) 
how  many  hills  to  the  acre  ?  (b)  If  the  corn  is  cut  and 
shocked,  putting  "  8  hills  square "  in  a  shock,  how  many 
shocks  to  the  acre  ?  (c)  If  there  are  "16  hills  square  "  in 
each  shock,  how  many  shocks  to  the  acre  ? 

99.  If  it  is  worth  $1.00  a  cord  to'  cut  "4-foot  wood" 
into  16-inch  pieces,  how  much  is  it  worth  to  cut  "8-foot 
wood  "  into  pieces  of  the  same  length  ? 

100.  An  agent  sold  1460  lb.  butter  at  23J^  a  pound.  If 
his  commission  for  selling  is  5%  and  he  paid  charges 
amoimting  to  $8.96,  how  much  should  he  remit  to  the, 
owner  of  the  butter  ? 

101.  A  lot  50  feet  wide  and  120  feet  "deep"  (long)  was 
sold  for  $450.     This  is  equivalent  to  what  price  per  acre  ? 

102.  Weight  of  wagon  and  hay,  4750  lb.;  weight  of 
wagon  1620  lb.  How  much  is  the  hay  worth  at  $12.50  per 
ton? 

103.  How  many  acres  in  5|-  miles  of  4-rod  road  ? 

104.  The  specific  gravity  of  ice  is  .92.  (a)  How  much 
does  a  cubic  foot  of  ice  weigh  ?  (b)  How  many  tons  of  ice, 
if  packed  solid,  can  be  stored  in  a  building  12  feet  square, 
the  ice  to  be  8  feet  deep  ? 

105.  How  many  tons  in  an  acre  of  ice  1 5  inches  thick  ? 

106.  On  the  first  day  of  May  the  water-meter  at  the  Illi- 
nois Institution  for  the  Blind  stood  at  375,400  (cu.  ft.);  on 


PART    III.  431 

the  first  day  of  June,  the  reading  was  477,700.  Eegarding 
each  cubic  foot  as  7i  gallons,  (a)  how  many  gallons  of  water 
were  used  in  May?  (b)  What  was  the  amount  of  the  bill 
for  water  if  the  price  was  12^  per  thousand  gallons,  with  a 
discount  of  1 6|  per  cent  ? 

107.  At  $1000  an  acre,  find  the  value  of  a  strip  of  land 
4  feet  by  125  feet. 

108.  If  the  specific  gravity  of  iron  is  7^,  how  many  cubic 
feet  in  1  ton  of  iron  ? 

109.  A  merchant  marked  goods  25%  above  the  cost;  he 
sold  them  at  25%  below  the  marked  price.  What  per  cent 
did  he  lose? 

110.  The  cube  of  a  number  divided  by  the  number  equals 
1764.     What  is  the  number? 

111.  What  is  the  edge  of  a  cube  whose  entire  surface  is 
6144  sq.  inches? 

112.  Seven  and  one  half  feet  are  what  part  of  a  rod? 

113.  Peter  has  12|^%  more  money  than  Paul;  together 
they  have  $6.97.     How  much  money  has  each? 

114.  Change  the  following  to  a  common  fraction  in  its 
lowest  terms:  •27y3y. 

115.  Answer  the  following  at  sight: 

(a)  Interest  of  $48  for  2  mo.  at  6%. 

(b)  Interest  of  $375  for  2  mo.  at  6%. 

(c)  Interest  of  $240  for  4  mo.  at  6%. 

(d)  Interest  of  $330  for  4  mo.-  at  6%. 

116.  Answer  the  following  at  sight: 

(a)  Divide  125  by  .5. 

(b)  Divide  125  by  .05. 

(c)  Divide  12.5  by  .5.       . 

(d)  Divide  .125  by  5. 


432  COMPLETE    ARITHMETIC. 

SET   I. 

COOK  COUNTY,  ILLINOIS. 

Eighth  Grade  Examination  for  County  Superintendent's  Diploma. 

June,  1897.     O.  T.  Bright,  Supt. 

Time,  9:30  to  12. 

1.  What  is  the  ratio  of  (a)  .2%  to  2%  ?  (b)  5  -f-  .5  =  ? 
(c).05-^5-?  (d).5-^.05  =  ?  (e).005-v-.5  =  ?  (f)  .05 -4- 
.005  =  ? 

2.  A  girl  spelled  95%  of  60  words.  How  many  words 
did  she  miss  ? 

3.  When  a  vessel  sails  160  miles  a  day,  she  completes 
her  voyage  in  14  days.  In  what  time  would  she  complete 
it  if  she  sailed  196  miles  a  day  ? 

4.  A  three-inch  cube  was  painted  on  all  sides.  It  was 
then  cut  into  inch  cubes,  (a)  How  many  of  the  inch  cubes 
were  painted  on  three  sides  ?  (b)  How  many  on  two  sides  ? 
(c)  How  many  on  one  side  ?  (d).  How  many  were  not 
painted  at  all  ? 

5.  Fill  the  blanks  in  the  following : 

A  boy  had  a  fish  pole  15  feet  long.  A  piece  equal  to  20% 
of  its  length  was  broken  off  while  catching  fish. 

(a)  The  part  remaining  was  %  of  the  whole  pole. 

(b)  The  part  broken  off  was  %  of  the  part  remaining. 

(c)  The  part  remaining  was  %  of  the  part  broken  off. 

6.  The  N.  E.  \  of  the  N.  W.  i  of  a  certain  section  of  land 
was  fenced  off  into  four  equal  fields,  (a)  What  is  the 
shortest  length  of  fence  necessary  ?  (b)  How  much  land  in 
each  field  ? 

7.  A  straight  pole  72  feet  high  is  broken  20  feet  from  the 
ground,  but  is  not  detached.  How  far  from  the  foot  will  the 
top  reach  ? 


PART   III.  433 

SET  11. 

NEW  HAVEN   (CONN.)   PUBLIC  SCHOOLS. 

Entrance  Examination  to  the  High  Schools. 

Fall,  1897.     C.  N.  Kendall,  Supt. 

(Answer  10  entire  questions.) 

1.  (a)  13  oz.  is  what  per  cent,  of  5  lb.  avoirdupois  ? 

(b)  A  man  added  18  cows  to  his  herd,  thereby  increasing 
the  number  25  per  cent.     How  many  cows  has  he  now  ? 

2.  If  lead  pencils  that  cost  3  cents  each  are  sold  for 
5  cents  each,  what  is  the  per  cent  of  profit  ? 

3.  Find  the  interest  on  a  note  for  $330  at  6  per  cent, 
given  August  3,  last  year,  and  due  to-day. 

4.  Write  a  negotiable,  interest-bearing,  promissory  note. 

5.  (a)  Divide  |  of  |  of  7|  by  3f . 

(b)  Subtract  8^?^  from  the  sum  of  5  J,  2|,  4:^^. 

6.  A  rectangular  field  is  86|^  rods  long  and  46.875  rods 
wide.  How  much  wheat  will  it  produce  at  the  rate  of  20 
bushels  per  acre  ? 

7.  A  rectangular  park,  the  sides  of  which  are  respectively 
45  rods  and  60  rods  long,  has  a  walk  crossing  it  from  corner 
to  corner.     How  long  is  the  walk  ? 

8.  If  f  of  9  bushels  of  wheat  cost  $13^,  what  will  |  of 
a  bushel  cost  ? 

9.  If  hay  sells  for  $14  a  ton  at  a  loss  of  12  J  per  cent, 
what  must  it  sell  for  to  gain  1 5  per  cent  ? 

10.  How  many  pounds  of  cotton  at  7-^  cents  a  pound  can  a 
broker  buy  for  $9,225,  and  retain  his  commission  of  2^  per 
cent  ? 

11.  Find  the  proceeds  of  a  3  months'  note  for  $500  dis- 
counted at  a  bank  at  6  per  cent. 

12.  If  a  building  20  feet  high  casts  a  shadow  of  6  feet, 
what  length  of  shadow  will  a  church  spire  114  feet  high  cast  1 


434  COMPLETE    ARITHMETIC. 

SET   III. 

COOK  COUNTY,  ILLINOIS. 

Applicants  for  Teachers'  Second  Grade  Certificates. 

June,  1898.     O.  T.  Bright,  Supt. 

"  Mental  Arithmetic."     Time,  20  minutes. 

Fill  the  blanks : 

1.  By  a  sale  of  goods  I  lost  12i%.    The  cost  was % 

of  the  selling  price. 

2.  The  circumference  of  a   3-in.  circle  is  %  of  its 

radius. 

3.  A  five-inch  square  is  %  greater  than  a  four-inch 

square. 

4.  I  buy  apples  four  for  three  cents  and  sell  them  three 
for  four  cents.     I  gain  per  cent. 

5.  A  and  B  are  78  miles  apart  and  walk  toward  each 
other ;  A  walks  3  miles  an  hour  and  B  3^  miles  an  hour. 
When  they  meet  B  will  have  walked  miles. 

6.  A  stick  of  timber  15  inches  square  and  32  feet  long 
contains  board  feet. 

7.  A  horse  trots  23f  miles  in  2  J  hours.  His  rate  per 
hour  is  miles. 

8.  $154  was  divided  among  A,  B,  C,  and  D,  in  the  pro- 
portion of  ^,  ^,  ^,  ^.     C  got  dollars. 

9.  A  school" contains  50  pupils: 
Monday,  3  were  absent  in  the  forenoon, 
Tuesday,  2  were  absent  all  day, 
Wednesday  3  absent  forenoon  and  2  afternoon, 
Thursday,  4  absent  all  day, 

Friday,  all  present 

The  per  cent  of  attendance  for  the  week  was  . 

10.  64  gal.  of  wine  and  1 6  gal.  of  water  were  mixed.    One 
pint  of  the  mixture  contained  of  a  gal.  of  water. 


PART  III.  435 

SET  IV. 

COOK  COUNTY,  ILLINOIS. 

Applicants  for  Teachers'  Second  Grade  Certificate. 

August,  1895.     O.  T.  Bright,  Supt. 

Time,  60  minutes. 

1.  If  a  quarter  section  of  land  has  fenced  within  it  the 
largest  possible  circular  lot,  how  many  acres  of  the  quarter 
section  will  remain  outside  of  the  circle  ? 

2.  Find  the  value  of  the  following  lumber  at  $15  per  M.: 

20  pieces  2x4,  18  ft.  long. 

20  pieces  4x4,  12  ft.  long. 

20  pieces  3x10,  16  ft.  long. 

45  16  ft.  stock  boards,  15  inches  wide. 

3.  (a)  What  sum  invested  in  8%  bonds  at  33^%  premium 
will  yield  an  annual  income  of  $1200  ? 

(b)  What  if  the  bonds  were  33^  discount? 

4.  Find  the  value  of  a  piece  of  land  20  ft.  x40  rods  at 
$1000  per  acre. 

5.  What  is  the  ratio  of  3|-  to  |  ?     Answer  in  per  cent. 

SET  V. 

Examination  Department  of  the  University  of  the  State  of 

New  York,  January,  1898. 

100  credits.     Necessary  to  pass,  75. 

Time,  9:15  a.  m.  to  12:15  p.  m. 

Answer  the  first  five  questions  and  five  of  the  others,  but  no  more. 

If  more  than  five    of  the  others  are  answered  only  the  first  five 

answers  will  be  considered.      Give  all   operations  (except   mental 

ones)  necessary  to  find  results.     Reduce  each  result  to  its  simplest 

form  and  mark  it  ^ns.    Each  complete  answer  will  receive  10  credits. 

1.  Define  numerator,  denominator,  divisor,  factor,  pro- 
portion. 

2.  Find  the  weight  in  kilograms  of  a  stone  1  meter  square 


436  COMPLETE    ARITHMETIC. 

and  .4  of  a  meter  thick,  assuming  that  the  stone  is  2|  times 
as  heavy  as  water. 

o    c-      -,•.    3  X  4  X  4.2 

3.  Simphfy      ,^ 

4.  Find  the  interest  on  $375  at  4^%  from  July  1,  last 
year,  to  the  present  time. 

5.  Multiply  65.15  by  3.14159  and  divide  the  result  by 
57.296,  finding  a  result  correct  to  three  decimal  places. 

6.  Find  the  cost  at  $50  an  acre  of  a  rectangular  field 
1650  feet  long  and  825  feet  wide. 

7.  Find  the  time  required  to  fill  a  cistern  8  feet  square 
and  5  feet  deep  by  a  pipe  which  admits  water  at  the  rate  of 
1  quart  a  second. 

8.  Make  a  receipted  bill  of  the  following :  J.  L.  Eobbins 
&  Co.  sold  this  day  to  Samuel  Jones  8  yards  of  cloth  at  37|- 
cents,  24  yards  of  calico  at  8^  cents,  1  dozen  handkerchiefs 
at  12|-  cents  each,  and  3  dozen  towels  at  $2.50  a  dozen. 

9.  Find  the  cost  of  four  sticks  of  timber,  each  8  inches  by 
10  inches  and  30  feet  long,  at  $15  a  1000  feet  board  measure. 

10.  Find  the  least  common  multiple  of  153,  204,  and  510. 

11.  If  4%  bonds  to  the  amount  of  $8000  face  value  are 
bought  at  92^%,  find  the  cost  of  the  bonds,  and  the  rate  of 
income  on  the  investment. 

12.  If  3  men  can  do  a  piece  of  work  in  8  days  of  10 
hours  each,  how  many  men  will  be  required  to  do  the  same 
work  in  6  days  of  8  hours  each  ?     (Solve  by  proportion.) 

13.  By  selling  a  horse  for  $144,  a  profit  of  60  per  cent  is 
made;  find  the  cost  of  the  horse. 

14.  The  diameter  of  a  bicycle  wheel  is  28  inches;  find 
the  number  of  revolutions  it  makes  in  going  1  mile. 

15.  Find  the  square  root  of  7,  correct  to  three  decimal  places. 


PAKT  III.  437 


SET  VI. 


Examination  for  Admission  to  State  High  Schools,  Minnesota,  1898. 
Time  two  hours. 

Answer  any  six — no  more.  If  more  are  attempted  and  the  student 
does  not  designate  which  six  he  wishes  to  be  graded  upon,  the  first 
six  answers  will  be  taken. 

1.  a  (2)  .What  is  the  ratio  of  2  to  .90  ? 
b  (2)  What  is  the  ratio  of  |  to  ^V  ? 
c  (2)  What  is  the  ratio  of  ^\  to  |  ? 

d  (2)  What  is  the  ratio  of  90%  to  .09  ? 
e  (2)  What  is  the  ratio  of  70%  to  50%  ? 

2.  a  (5)  What  is  the  cubical  content  of  a  cellar  15  ft. 
wide,  20  ft.  long,  and  10  ft.  deep?  (In  the  solution  express 
all  operations  in  the  form  of  equations.) 

b  (1)  What  unit  (or  units)  of  measure  did  you  use  in  the 
example  ? 

c  (4)  Describe  the  unit  of  measure  used  in  measuring  boards. 

3.  (10)  A  merchant  sells  an  overcoat  for  S22 ;  a  suit  of 
clothes  for  $2  3  ;  a  hat  for  S5.  On  the  overcoat  he  makes  1 0  % 
of  the  cost ;  on  the  suit  15  %,  and  on  the  hat  25  %.  What  per 
cent  of  the  cost  of  the  goods  does  he  make  on  the  entire  sale  ? 

4.  a  (5)  A  man  bought  a  watch  and  a  chain  for  $70. 
One-half  of  the  cost  of  the  watch  equals  f  of  the  cost  of  the 
chain.     What  was  the  cost  of  each  ? 

b  (5)  Analyze. 

5.  (10)  The  rates  at  which  A,  B,  and  C  work  are  to  each 
other  as  2,  3,  and  4.  What  integers  wdll  indicate  the  time 
it  will  take  each  to  do  a  certain  piece  of  work  ? 

6.  (10)  How  long  a  rope  must  a  horse  have  in  order  that 
he  may  graze  over  an  acre  of  land,  if  he  be  tied  to  a  stake 
in  the  center  of  a  field  ? 


438  COMPLETE    ARITHMETIC. 

7.  (10)  B  buys  bank  stock  at  78  and  sells  it  at  84.  C 
buys  railroad  stock  at  70  and  sells  at  75.  Each  buys  the 
same  number  of  shares,  and  B  makes  $1000  more  than  C. 
How  much  money  did  B  invest? 

8.  (10)  Make  (5)  and  solve  (5)  a  problem  in  the  solution 
of  which  it  will  be  necessary  to  extract  the  square  root. 

9.  a  (5)  What  is  the  interest  on  $700  for  1  yr.  5  mo. 
and  10  da.  at  7%  per  annum? 

b  (5)  Analyze. 

10.  (10)  A  servant  is  engaged  for  a  year  for  $280  and  a 
suit  of  clothes.  He  leaves  at  the  end  of  six  months  and 
receives  $130  and  the  suit.  What  is  the  value  of  the  suit? 
(An  algebraic  solution  is  allowed  for  this  problem.) 

11.  a  (5)  How  long  will  it  take  $1560  at  5%  simple 
interest  to  gain  $426.83^? 

b  (5)  Analyze. 

SET  VII. 

Examination  for  State  Certificates.     lUinois,  1898. 
Time,  two  hours. 

1.  (a)  Every  fraction  is  a  ratio.     Explain. 

(b)  Every  integral  number  is  a  ratio.     Explain. 

2.  In  the  report  of  the  Committee  of  Ten  it  is  recom- 
mended that  "  the  course  in  arithmetic  be  at  the  same  time 
abridged  and  enriched." 

(a)  Tell  what  abridgment  you  regard  as  important. 

(b)  Tell  what  enrichment  you  consider  essential. 

3.  (a)  Tell  what  sense-magnitudes  you  prefer  to  use  in 
presenting  to  third  grade  pupils  the  subject  of  fractions, 
(b)  At  wliat  stage  of  the  work  do  you  think  sense-magni- 
tudes should  give  place  to  imaginative  magnitudes  ? 


PART   III.  439 

4.  When  and  to  what  extent  should  pupils  in  the  grades 
be  required  to  memorize  definitions  of  mathematical  terms  ? 
(b)  When  and  to  what  extent  should  pupils  be  required  to 
memorize  directions  for  performing  operations  ? 

5.  Mention  all  the  standard  linear  units  with  which  you 
are  familiar,  and  give  the  ratio  of  each  (either  exact  or 
approximate)  to  some  other  linear  unit. 

6.  What  is  the  weight  of  1000  feet  of  white  pine  boards 
(1  inch  in  thickness)  if  the  specific  gravity  of  the  boards 
is  .6? 

7.  The  foundation  of  my  house  is  32  feet  square  on  the 
outside.  The  house  is  20  feet  high  to  the  plates  and  the 
roof  has  the  usual  eave-projections.  Give  approximately  the 
number  of  barrels  (31i  gal.)  of  water  that  will  fall  upon  this 
roof  in  one  year,  if  the  rain-fall  is  34 J  inches. 

8.  Give  approximately  the  following  ratios: 

(a)  Of  the  circumference  to  the  diameter  of  a  circle. 

(b)  Of  the  diagonal  to  the  side  of  a  square. 

(c)  Of  the  area  of  a  circle  to  the  area  of  its  circumscribed 
square. 

(d)  Of  the  area  of  a  circle  to  the  area  of  its  inscribed 
square. 

9.  What  single  discount  is  equal  to  a  discount  of  45  per 
cent  and  "  5  lO's,"  i.  e.,  to  "45  and  10  and  10  and  10  and 
10  and  10.  off,"  from  the  list  price  ? 

10.  If  money  is  worth  6  per  cent  annual  interest  now  and 
prospectively,  what  is  the  actual  cash  value  of  a  note  of 
SI 000  running  two  years  and  drawing  5  per  cent  interest, 
payable  annually  ? 


440  COMPLETE    ARITHMETIC. 

VIII.  The  Bank  Test.* 

To  THE  Tkachp:k. — Below  are  figures  representing  51  sums  of 
money.  Procure  51  blank  checks  and  cause  them  to  be  filled,  using 
the  sums  here  given.  Draw  one  check  from  the  51  checks  and 
give  the  remaining  50  to  a  pupil  to  transcribe  the  sums  and  find  their 
amount.  When  the  pu^Dil  obtains  a  result  the  teacher  can  quickly 
determine  whether  it  is  correct  by  comparing  it  with  the  sum  of  the 
51  checks,  less  the  sum  named  on  the  check  drawn  out.  Before  the 
checks  are  given  to  the  second  pupil,  the  check  removed  should  be 
replaced  and  another  withdrawn.  Thus,  although  each  pupil  should 
obtain  a  result  differing  from  that  obtained  by  the  pupil  preceding 
him,  its  accuracy  can  be  quickly  tested  by  the  teacher. 

To  THE  Pupil. — Can  you,  on  first  trial,  transcribe  the  sums  named 
on  50  checks  and  find  the  amount  accurately  in  30  minutes  ? 

$324.56   $565.60   $123.20   $75.00 
$234.50   $525.40   $312.95   $190.35 
$46.45    $112.00   $86.50    $250.00 
$325.00   $86.74    $91.23    $50.00 
$302.26   $59.29    $12.65    $8.25 
$7.75     $875.00   $1.50     $431.05 
$201.45   $34.36    $85.40    $90.00 
$130.25   $212.24   $230.94   $642.45 
$71.20    $708.30   $60.00    $75.00 
$1250.25  $6.50     $500.00   $2324.45 
$9.10     $101.50   $36.09    $275.00 
$150.00   $2.50     $1008.60  $140.65 
$256.74   $987.84   $50.00 


*In  a  leading  bank  in  Chicago,  it  is  customary  to  test  applicants  for  positions  as 
accountants  by  placing  before  them  150  checks,  requiring  each  applicant  to  copy  the 
sums  named  on  the  checks  and  find  their  amount.  The  author  of  this  book  is 
informed  that  the  average  inexperienced  applicant  does  this  in  about  30  minutes, 
with  some  errors,  however,  both  in  transcribing  and  in  footing.  An  expert 
accountant  can  do  this  amount  of  work  accurately  in  6  minutes. 


PART    III.  441 

IX.    Curious  Comparisons. 

1.  If  a  pig  whose  girth  is  2  feet  weighs  50  lb.,  what  is 
the  weight  of  a  similarly  proportioned  pig  whose  girth  is  4  feet? 

2.  If  a  disk  of  dough  15  inches  in  diameter  is  sufficient 
for  20  doughnuts,  how  many  such  doughnuts  can  be  made 
from  a  disk  30  inches  in  diameter  ? 

3.  The  bore  of  a  1 0-inch  gun  is  how  many  times  as  large 
as  the  bore  of  a  2 -inch  gun  ? 

4.  The  ball  of  a  10-inch  gun  is  how  many  times  as  large 
as  the  ball  of  a  2 -inch  gun  ? 

5.  A  square,  a  pentagon,  a  hexagon,  an  octagon,  and  a 
circle  have  equal  perimeters,  (a)  Which  has  the  greatest 
area  ?     (b)  Which  has  the  least  area  ? 

6.  The  capacity  of  a  cistern  6  feet  in  diameter  and  6  feet 
deep  is  about  40  barrels.  What  is  the  capacity  of  a  cistern 
12  feet  in  diameter  and  12  feet  deep  ? 

7.  A  2|-inch  pipe  is  how  many  times  as  large  as  a  1-inch 
pipe  ? 

8.  If  a  man  6  feet  tall  w^eighs  190  lb.,  how  much  would 
a  similarly  proportioned  giant  12  feet  tall  weigh  ? 

9.  In  a  certain  orchard  the  trees  are  15  feet  apart  each 
way  and  there  are  800  trees.  How  many  trees  in  an  orchard 
of  equal  size,  the  trees  being  30  feet  apart  each  way  ? 

10.  A  ball  of  yarn  3  inches  in  diameter  is  sufficient  for 
one  mitten.  How  many  mittens  can  be  made  from  a  ball  6 
inches  in  diameter  ? 

11.  A  grindstone  was  originally  30  inches  in  diameter. 
It  has  been  worn  until  it  is  but  15  inches  in  diameter.  What 
part  of  the  stone  has  been  worn  away  ? 

12.  A  square  and  an  oblong  have  equal  areas.  Which 
has  the  greater  perimeter  ? 


442  COMPLETE    ARITHMETIC. 

X.    Puzzling  Problems. 

1.  If  a  person  traveling  as  expeditiously  as  possible  from 
Boston  to  San  Francisco,  should  mail  a  letter  to  his  friend 
in  Boston  every  day  at  noon,  how  often  would  the  letters  be 
received  in  Boston  ? 

2.  If  a  man  and  a  boy,  the  boy  doing  exactly  one-half  as 
much  work  as  the  man,  can  hoe  one  and  one-half  acres  of 
corn  in  one  and  one-half  days,  how  many  acres  can  6  men 
hoe  in  6  days  ? 

3.  John  and  James  sold  apples  together.  The  first  day 
they  sold  60  apples  at  the  rate  of  5  apples  for  2  cents,  and  re- 
ceived 24  cents.  The  second  day  they  divided  the  apples.  John 
took  30  of  the  larger  apples  and  sold  them  at  the  rate  of  2  for 
1  cent.  James  took  the  remaining  30  apples  and  sold  them  at 
the  rate  of  3  for  1  cent.  They  received  25  cents.  Why  did 
they  receive  one  cent  more  the  second  day  than  the  first  ? 

4.  A  pile  of  four-foot  wood  stands  upon  a  hill-side.  The 
pile  is  8  feet  long  (measured  on  the  ground),  and  4  feet  high 
(measured  vertically).     Does  the  pile  contain  one  cord  ? 

5.  A  man  had  shingles  enough  to  cover  his  house  if  he 
laid  them  4  inches  to  the  weather.  He  laid  them  4i  inches 
to  the  weather.  What  part  of  the  shingles  provided 
remained  ?     Explain. 

6.  If  on  a  line  of  railroad  connecting  Chicago  and  San 
Francisco  one  passenger  train  leaves  Chicago  daily  at  6  o'clock 
a.  m.,  and  makes  the  journey  to  San  Francisco  in  exactly  five 
days,  and  one  train  leaves  San  Francisco  daily  at  6  o'clock 
p.  m.,  and  makes  the  journey  to  Chicago  in  exactly  five  days, 
(a)  a  person  taking  the  train  at  Chicago  will  meet  how  many 
passenger  trains  while  going  to  San  Francisco  ?  (b)  How 
many  trains  of  passenger  cars  required  to  equip  the  road  ? 


EXPLANATOEY   NOTES. 
Note  1 .     The  forty-five  primary  facts  of  addition  are  as  follows : 


1 

2 

2 

3 

3 

4 

3 

4 

5 

4 

5 

6 

4 

5 

6 

1 

1 

2 

1 

2 

1 

3 

o 

1 

3 

2 

1 

4 

3 

2 

2 

3 

4 

4 

5 

5 

6 

6 

G 

7 

7 

7 

8 

8 

8 

7 

o 

6 

7 

8 

5 

6 

7 

8 

9 

6 

7 

8 

9 

6 

1 

4 

3 

2 

1 

5 

4 

3 

2 

1 

5 

4 

3 

2 

6 

8 

9 

9 

9 

9 

10 

10 

10 

10 

10 

11 

11 

11 

11 

12 

7 

8 

9 

7 

8 

9 

7 

8 

9 

8 

9 

8 

9 

9 

9 

5 

4 

3 

6 

5 

4 

7 

6 

5 

7 

6 

8 

7 

8 

9 

The  nine  facts  in  full-faced  type  should  receive  special  attention. 
Pupils  seldom  fail  to  memorize  the  other  thirty-six  facts. 

Note  2.  There  are  eighty-one  primary  facts  of  subtraction;  that 
is,  two  for  every  primary  fact  of  addition  except  the  1st,  3rd,  7th, 
13th,  21st,  30th,  37th,  42nd,  and  45th.  The  facts  of  subtraction 
should  be  learned  while  learning  the  facts  of  addition.  If  a  pupil 
really  knows  that  8  and  9  equal  17,  he  knows  also  that  17  less  8  =  9, 
and  17  less  9  =  8. 

Note  3.  When  the  sign  of  multiplication  is  followed  by  a  frac- 
tion, it  indicates  that  a  certain  part  of  the  number  preceding  the 
sign  is  to  be  repeated  as  many  times  as  there  are  units  in  the 
numerator  of  the  fraction  following  the  sign;  thus,  12  x  I,  means, 
that  1  fourth  of  12  is  to  be  repeated  3  times;  50  X  -5,  means,  that 
1  tenth  of  50  is  to  be  repeated  5  times. 

Note  4.  This  sign,  X,  is  sometimes  so  used  that  it  means  times, 
thus,  3  X  <156,  must  be  read,  three  times  six  dollars.  3x6,  may  be 
read,  three  multiplied  by  six  or  three  times  six.     As  employed  in  this 

443 


444 


COMPLETE    ARITHMETIC. 


book,  the  sign  never  means  times.  Instead  of  3  x  $6,  the  author  pre- 
fers $6  X  3.  It  is  believed  that  the  restriction  of  this  sign  to  one 
use  and  to  one  meaning,  at  least  in  the  first  years  of  arithmetical 
study,  will  promote  clearness  of  thought  and  accuracy  in  expression. 

Note  5.  Without  danger  of  ambiguity,  the  sign,  X,  is  sometimes 
used  in  this  book  and  elsewhere  in  place  of  the  word  by;  thus,  1  pc. 
of  2  X  4,  12  (to  be  read,  1  pc.  of  2  hy  4,  12)  means,  a  piece  of  lum- 
ber 2  inches  thick,  4  inches  wide,  and  12  feet  long. 

Note  6.  Besides  those  problems  in  which  either  the  multipli- 
cand or  the  multiplier  is  1,  and  which  require  no  eifort  on  the  part 
of  the  pupil  beyond  learning  to  count,  there  are  sixty-four  primary 
facts  of  multiplication  that  must  be  perfectly  memorized  before  the 
pupil  can  acquire  facility  in  the  process.     They  are  as  follows : 

2  times  2=4      3  times  2=6      4  times  2=8      5  times  2  =  10 


2   « 

3=6 

3 

3=9 

4 

3  =12 

5 

3  =  15 

2   " 

4=8 

3 

4  =  12 

4 

4  =  16 

5 

4=20 

2   " 

.5  =  10 

3 

5  =  15 

4 

5  =  20 

5 

5  =  25 

2   " 

6  =  12 

3 

6  =  18 

4 

6  =  24 

5 

6  =30 

2   " 

7  =  14 

3 

7  =  21 

4 

7  =28 

5 

7  =  35 

2   " 

8  =  16 

3 

8  =  24 

4 

8  =  32 

5 

8  =  40 

2   " 

9  =  18 

3 

9  ==27 

4 

9  =36 

5 

9  =45 

6  times  2  =  12 

7  times 

2  =  14 

8  times  2  =  16 

9  times  2  =18 

6  " 

3  =  18 

7 

3  =  21 

8 

3  =  24 

9 

3  =  27 

6   " 

4  =  24 

7 

4  =  28 

8 

4  =  32 

9 

4  =  36 

6   " 

5  =  30 

7 

5  =  35 

8 

5  =  40 

9 

5  =  45 

6   " 

6  =  36 

7 

6  =42 

8 

6  =  48 

9 

6  =  54 

6   ^' 

7  =  42 

7 

7  =49 

8 

7  =  56 

9 

7  =  63 

6   " 

8  =  48 

7 

8  =  56 

8 

8  =  64 

9 

8  =  72 

6   " 

9  =  54 

7 

9  =  63 

8 

9  =  72 

9 

9  =  81 

Although  a  knowledge  of  the  "elevens"  and  "twelves"  of  the 
table  as  it  is  usually  given  is  convenient  and  helpful,  it  will  be 
observed  that  it  is  not  a  necessity  in  the  process  of  multiplication. 
The  facts  given  above  include  all  that  are  essentially  fundamental. 


PART   III.  445 

Note  7.  The  sign  -7-,  which  is  read  divided  by,  has  two  meanings 
in  concrete  problems,  which  correspond  to  the  two  cases  in  division. 
In  one  case  it  means,  find  how  many  times  the  divisor  is  contained  in 
the  dividend;  in  the  other  case  it  mea^ns,  find  one  of  a  certain  number 
of  equal  parts  into  which  the  dividend  is  supposed  to  be  divided.  In 
each  case  there  is  division  into  equal  parts.  In  the  first  case,  the 
quotient  tells  the  number  of  parts.  In  the  second  case,  the  quotient 
tells  the  size  of  one  part. 

$18  -T-  $2,  means,  ,y?n(Z  how  many  times  $2  are  contained  in  $18. 

$18  -f-  2,  means,  find  1  half  of  $18.      (See  foot-note,  p.  192.) 

Note  8.  There  are,  in  a  sense,  128  primary  facts  of  division, — 
two  for  each  one  of  the  sixty-four  facts  of  multiplication.  These 
facts  are  so  closely  related  to  the  facts  of  multiplication  that  they 
should  be  learned  in  connection  with  the  multiplication  table.  If  a 
child  really  perceives  that  five  fours  (  :  :  :  :  : :  : :  :  : )  are  20,  he  will 
also  Know  that  4  is  contained  lu  20  five  times,  and  that  1  fifth  of 
20  is  4. 

Note  9.  If  from  a  square  piece  of  paper,  the  largest  possible 
circle  be  cut,  a  little  less  than  \  of  the  paper  will  be  cut  away. 
Hence  a  circle  is  a  little  more  than  |  (.78+)  of  its  circumscribed 
square. 

Observe  that  the  diameter  of  a  circle  is  equal  to  the  side  of  its 
circumscribed  square. 

Note  10.  If  from  a  cube  of  wood,  the  largest  possible  sphere  be 
cut,  a  little  less  than  \  of  the  wood  will  be  cut  away.  Hence  a 
sphere  is  a  little  more  than  \  (.52  +)  of  its  circumscribed  cube. 

Observe  that  the  diameter  of  a  sphere  is  equal  to  the  edge  of  its 
circumscribed  cube. 

Note  11.  A  piece  of  board  1  inch  wide,  1  inch  thick,  and  12 
feet  long,  is  1  ft.  of  lumber.  Hence  the  number  of  feet  of  lumber 
in  any  12-foot  stick,  is  equal  to  the  number  of  square  inches  in  its 
cross-section. 

Note  12.  If  from  a  square  right  prism  of  wood  the  largest  possi- 
ble cylinder  be  cut,  a  little  less  than  \  of  the  wood  will  be  cut  away. 


446  COMPLETE    ARITHMETIC. 

Hence  a  cylinder  is  a  little  more  than  |  (.78  +  )  of  its  circumscribed 
square  right  prism. 

Note  13.  If  from  a  cylinder  of  wood  whose  diameter  equals  its 
altitude  the  largest  possible  sphere  be  cut,  exactly  ^  of  the  wood  will 
be  cut  away.  Hence  a  sphere  is  exactly  |  of  a  cylinder  whose 
diameter  and  altitude  are  each  equal  to  the  diameter  of  the  sphere. 

Note  14.  If  from  a  cylinder  of  wood  the  largest  possible  cone 
be  cut,  exactly  |  of  the  wood  will  be  cut  away.  Hence  a  cone  is 
exactly  ^  of  a  cylinder  of  equal  diameter  and  altitude. 

Note  15.  If  from  a  square  right  prism  of  wood  the  largest 
possible  pyramid  be  cut,  exactly  |  of  the  wood  will  be  cut  away. 
Hence  a  square  pyramid  is  exactly  -^-  of  a  square  right  prism  of 
equal  base  and  altitude. 

Note  16.  The  specific  gravity  of  a  liquid  or  solid  is  the  ratio  of 
its  weight  to  the  weight  of  an  equal  bulk  of  pure  water. 


PROTRACTOR. 


Carefully  paste  this  sheet  upon  card-board;  then  cut 
out  the  protractor  with  a  sharp  knife  and  preserve  it 
for  use  in  measuring  and  in  constructing  angles.  See 
pages  239,  249,  259,  etc. 


447 


VB  35822 


541^20 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


